Tutorial 4 - Li et al., 2022

1.4 Model the Data Generation Process

First let's define the functions

function d_Inequality = inequality(a1, b1, a2, b2)
d_Inequality = abs(a2 - b2) - abs(a1 - b1);
end
 
function d_lossAdvantaged = harm(a0, b0, a1, b1, a2, b2)
initial = [a0, b0];
choice1 = [a1, b1];
choice2 = [a2, b2];
if a0 == b0
    advantaged = 1;
else
    [~, advantaged] = max(initial);
end
d_lossAdvantaged = (initial(advantaged) - choice1(advantaged)) - (initial(advantaged) - choice2(advantaged));
end
 
function rankReverseDiff = rankReverse(a0, b0, a1, b1, a2, b2)
if a0 == b0
    rankReverseDiff = 0;
    return;
end
d_initial = a0 - b0;
d_choice1 = a1 - b1;
d_choice2 = a2 - b2;
choice1Reversed = 0;
choice2Reversed = 0;
if d_initial > 0
    if d_choice1 < 0, choice1Reversed = 1; end
    if d_choice2 < 0, choice2Reversed = 1; end
elseif d_initial < 0
    if d_choice1 > 0, choice1Reversed = 1; end
    if d_choice2 > 0, choice2Reversed = 1; end
end
rankReverseDiff = choice1Reversed - choice2Reversed;
end

Now let's check and see if they do what we want. Let's make an example trial:

example = table(20, 0, 8, 12, 15, 5, 'VariableNames', ["a0", "b0", "a1", "b1", "a2", "b2"])
example = 1×6 table 
 a0b0a1b1a2b2
1200812155

	a0	b0	a1	b1	a2	b2
1	20	0	8	12	15	5

Now get the outputs based on this example

inequality(example.a1(1), example.b1(1), example.a2(1), example.b2(1))
ans = 6
harm(example.a0(1), example.b0(1), example.a1(1), example.b1(1), example.a2(1), example.b2(1))
ans = 7
rankReverse(example.a0(1), example.b0(1), example.a1(1), example.b1(1), example.a2(1), example.b2(1))
ans = 1

1.5 Simulating Data

Now let's preallocate and define functions, triallist, and parameters

a0 = randi([10,20], 100, 1);
b0 = 20 - a0;
trialList = table(a0, b0);
clear a0 b0
 
for i = 1:length(trialList.a0)
    trialList.a1(i) = randi([5, trialList.a0(i)]);
end
trialList.b1 = 20 - trialList.a1;
 
for i = 1:length(trialList.a0)
    trialList.a2(i) = randi([5, trialList.a0(i)]);
    if trialList.a2(i) == trialList.a1(i)
        trialList.a2(i) = 10;
    end
end
trialList.b2 = 20 - trialList.a2;
trialList = [trialList; trialList(:, [2, 1, 4, 3, 6, 5])]
trialList = 200×6 table 
 a0b0a1b1a2b2
1182713146
21911641010
31197131010
42001371010
5164614911
6119911119
7137713128
8164128119
9200164146
10200164146
11119812614
12200614911
13200812128
14155155713
⋮
 
function util = utility(pars, IVs)
IVs = double(IVs);
a0 = IVs(1); b0 = IVs(2); a1 = IVs(3); b1 = IVs(4); a2 = IVs(5); b2 = IVs(6);
alpha = pars(1); delta = pars(2); rho = pars(3);
util = (alpha * inequality(a1, b1, a2, b2)) - (delta * harm(a0, b0, a1, b1, a2, b2)) - (rho * rankReverse(a0, b0, a1, b1, a2, b2));
end
 
function prob = probability(pars, utilitydiff)
beta = pars(end-2); epsilon = pars(end-1); gamma = pars(end);
prob = 1 / (1 + exp(-(beta * utilitydiff)));
prob = prob * (1 - 2 * epsilon) + epsilon + gamma * (2 * epsilon);
prob = max(min(prob, 0.9999999999), 0.00000000001);
end
 
freeParameters = struct();
mainpars = (0:3)./2;
bet = (0:5:5);
eps = (0:0.1:0.5);
gam = (-0.5:0.5:0);
for i = 1:length(mainpars)
    for j = 1:length(mainpars)
        for k = 1:length(mainpars)
            for l = 1:length(bet)
                for m = 1:length(eps)
                    for n = 1:length(gam)
                        freeParameters(i, j, k, l, m, n).alpha = mainpars(i) + rand(1,1)*0.5;
                        freeParameters(i, j, k, l, m, n).delta = mainpars(j) + rand(1,1)*0.5;
                        freeParameters(i, j, k, l, m, n).rho = mainpars(k) + rand(1,1)*0.5;
                        freeParameters(i, j, k, l, m, n).beta = bet(l).*rand(1,1)*5;
                        freeParameters(i, j, k, l, m, n).epsilon = eps(m);
                        freeParameters(i, j, k, l, m, n).gamma = gam(n) + rand(1, 1)*0.5;
                    end
                end    
            end
        end
    end
end 
 
function pred = generatePredictions(parameters, df)
pred = zeros(size(df, 1), 1);
for i = 1:size(df, 1)
    thisTrialIVs = table2array(df(i, :));
    utilityDiff = utility(parameters, thisTrialIVs);
    pred(i) = probability(parameters, utilityDiff);
end
end

	a0	b0	a1	b1	a2	b2
1	18	2	7	13	14	6
2	19	1	16	4	10	10
3	11	9	7	13	10	10
4	20	0	13	7	10	10
5	16	4	6	14	9	11
6	11	9	9	11	11	9
7	13	7	7	13	12	8
8	16	4	12	8	11	9
9	20	0	16	4	14	6
10	20	0	16	4	14	6
11	11	9	8	12	6	14
12	20	0	6	14	9	11
13	20	0	8	12	12	8
14	15	5	15	5	7	13
⋮

Now that all of that's done, let's generate predictions

for i = 1:length(mainpars)
    for j = 1:length(mainpars)
        for k = 1:length(mainpars)
            for l = 1:length(bet)
                for m = 1:length(eps)
                    for n = 1:length(gam)
                        pars = [freeParameters(i, j, k, l, m, n).alpha, freeParameters(i, j, k, l, m, n).delta, freeParameters(i, j, k, l, m, n).rho, freeParameters(i, j, k, l, m, n).beta, freeParameters(i, j, k, l, m, n).epsilon, freeParameters(i, j, k, l, m, n).gamma];
                        freeParameters(i, j, k, l, m, n).predictions = generatePredictions(pars, trialList);
                    end
                end    
            end
        end
    end
end 

1.6 Compare Recovered Parameters

Let's write the objective function

function obj_val = obj_function(params, df, optimMethod)
if nargin < 3
    optimMethod = 'MLE';
end
Prob1 = generatePredictions(params, df);
Chose1 = table2array(df(:, 7));
if strcmp(optimMethod, 'OLS')
    obj_val = sum((Chose1 - Prob1) .^ 2);
else % MLE
    obj_val = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(Prob1));
end
end

Now we can set up the optimizer (we'll suppress the output of successful completion to avoid inundating the output)

initial_params = [1, 1, 1, 4, 0.25, 0];
lower_bounds = [0, 0, 0, 0, 0, -0.5];
upper_bounds = [2, 2, 2, 10, 0.5, 0.5];
 
function result = optimize(obj, initial_params, lower_bounds, upper_bounds, df)
 
try
    result = fmincon(@(x) obj(x, df, 'OLS'), initial_params, [], [], [], [], lower_bounds, upper_bounds, [], optimoptions('fmincon', 'Display', 'off'));
catch
    result = fmincon(@(x) obj(x, df, 'MLE'), initial_params, [], [], [], [], lower_bounds, upper_bounds, [], optimoptions('fmincon', 'Display', 'off'));
end
end

And this lets us recover the free parameters

for i = 1:length(mainpars)
    for j = 1:length(mainpars)
        for k = 1:length(mainpars)
            for l = 1:length(bet)
                for m = 1:length(eps)
                    for n = 1:length(gam)
                        trialList.Chose1(:) = round(freeParameters(i, j, k, l, m, n).predictions .* 1000) > randi([1, 1000], length(freeParameters(i, j, k, l, m, n).predictions), 1);
                        result = optimize(@obj_function, initial_params, lower_bounds, upper_bounds, trialList);
                        freeParameters(i, j, k, l, m, n).alphaRecovered = result(1);
                        freeParameters(i, j, k, l, m, n).deltaRecovered = result(2);
                        freeParameters(i, j, k, l, m, n).rhoRecovered = result(3);
                        freeParameters(i, j, k, l, m, n).betaRecovered = result(4);
                        freeParameters(i, j, k, l, m, n).epsilonRecovered = result(5);
                        freeParameters(i, j, k, l, m, n).gammaRecovered = result(6);
                    end
                end    
            end
        end
    end
end 

Let's extract values to format in a plottable way

totalIterations = length(mainpars)^3 * length(bet) * length(eps) * length(gam);
parsPlot.alpha = NaN(totalIterations, 1);
parsPlot.alphaRecovered = NaN(totalIterations, 1);
parsPlot.delta = NaN(totalIterations, 1);
parsPlot.deltaRecovered = NaN(totalIterations, 1);
parsPlot.rho = NaN(totalIterations, 1);
parsPlot.rhoRecovered = NaN(totalIterations, 1);
parsPlot.beta = NaN(totalIterations, 1);
parsPlot.betaRecovered = NaN(totalIterations, 1);
parsPlot.epsilon = NaN(totalIterations, 1);
parsPlot.epsilonRecovered = NaN(totalIterations, 1);
parsPlot.gamma = NaN(totalIterations, 1);
parsPlot.gammaRecovered = NaN(totalIterations, 1);
for i = 1:length(mainpars)
    for j = 1:length(mainpars)
        for k = 1:length(mainpars)
            for l = 1:length(bet)
                for m = 1:length(eps)
                    for n = 1:length(gam)
                        parsPlot.alpha(end+1) = freeParameters(i, j, k, l, m, n).alpha;
                        parsPlot.alphaRecovered(end+1) = freeParameters(i, j, k, l, m, n).alphaRecovered;
                        parsPlot.delta(end+1) = freeParameters(i, j, k, l, m, n).delta;
                        parsPlot.deltaRecovered(end+1) = freeParameters(i, j, k, l, m, n).deltaRecovered;
                        parsPlot.rho(end+1) = freeParameters(i, j, k, l, m, n).rho;
                        parsPlot.rhoRecovered(end+1) = freeParameters(i, j, k, l, m, n).rhoRecovered;
                        parsPlot.beta(end+1) = freeParameters(i, j, k, l, m, n).beta;
                        parsPlot.betaRecovered(end+1) = freeParameters(i, j, k, l, m, n).betaRecovered;
                        parsPlot.epsilon(end+1) = freeParameters(i, j, k, l, m, n).epsilon;
                        parsPlot.epsilonRecovered(end+1) = freeParameters(i, j, k, l, m, n).epsilonRecovered;
                        parsPlot.gamma(end+1) = freeParameters(i, j, k, l, m, n).gamma;
                        parsPlot.gammaRecovered(end+1) = freeParameters(i, j, k, l, m, n).gammaRecovered;
                    end
                end    
            end
        end
    end    
end 

So we can now verify that we can reliably recover them using plots

parsPlot.Epsilon = categorical(parsPlot.epsilon);

figure;

scatter(parsPlot.alpha, parsPlot.alphaRecovered, [], parsPlot.Epsilon);

hold on; lsline; plot(xlim, ylim, '--k'); hold off;

title('Alpha'); xlabel('Actual'); ylabel('Recovered');

figure;

scatter(parsPlot.delta, parsPlot.deltaRecovered, [], parsPlot.Epsilon);

hold on; lsline; plot(xlim, ylim, '--k'); hold off;

title('Delta'); xlabel('Actual'); ylabel('Recovered');

figure;

scatter(parsPlot.rho, parsPlot.rhoRecovered, [], parsPlot.Epsilon);

hold on; lsline; plot(xlim, ylim, '--k'); hold off;

title('Rho'); xlabel('Actual'); ylabel('Recovered');

figure;

scatter(parsPlot.beta, parsPlot.betaRecovered, [], parsPlot.Epsilon);

hold on; lsline; plot(xlim, ylim, '--k'); hold off;

title('Beta'); xlabel('Actual'); ylabel('Recovered');

figure;

scatter(parsPlot.epsilon, parsPlot.epsilonRecovered);

hold on; lsline; plot(xlim, ylim, '--k'); hold off;

title('Epsilon'); xlabel('Actual'); ylabel('Recovered');

figure;

scatter(parsPlot.gamma, parsPlot.gammaRecovered, [], parsPlot.Epsilon);

hold on; plot(xlim, ylim, '--k'); hold off;

title('Gamma'); xlabel('Actual'); ylabel('Recovered');

These are a bit more difficult to visually interpret, but nothing looks particularly accurate. Let's take a closer look at situations where we should see good accuracy.

figure;

title('Minimal Noise')

subplot(3, 2, 1);

scatter(parsPlot.alpha(parsPlot.epsilon == 0 & parsPlot.beta > 5), parsPlot.alphaRecovered(parsPlot.epsilon == 0 & parsPlot.beta > 5));

hold on; plot(xlim, ylim, '--k'); hold off;

title('Alpha'); lsline; xlabel('Actual'); ylabel('Recovered');

subplot(3, 2, 2);

scatter(parsPlot.delta(parsPlot.epsilon == 0 & parsPlot.beta > 5), parsPlot.deltaRecovered(parsPlot.epsilon == 0 & parsPlot.beta > 5));

hold on; plot(xlim, ylim, '--k'); hold off;

title('Delta'); lsline; xlabel('Actual'); ylabel('Recovered');

subplot(3, 2, 3);

scatter(parsPlot.rho(parsPlot.epsilon == 0 & parsPlot.beta > 5), parsPlot.rhoRecovered(parsPlot.epsilon == 0 & parsPlot.beta > 5));

hold on; plot(xlim, ylim, '--k'); hold off;

title('Rho'); lsline; xlabel('Actual'); ylabel('Recovered');

subplot(3, 2, 4);

scatter(parsPlot.beta(parsPlot.epsilon == 0 & parsPlot.beta <=10), parsPlot.betaRecovered(parsPlot.epsilon == 0 & parsPlot.beta <=10));

hold on; plot(xlim, ylim, '--k'); hold off;

title('Beta'); lsline; xlabel('Actual'); ylabel('Recovered');

subplot(3, 2, 5);

scatter(parsPlot.epsilon(parsPlot.beta > 5), parsPlot.epsilonRecovered(parsPlot.beta > 5));

hold on; plot(xlim, ylim, '--k'); hold off;

title('Epsilon'); lsline; xlabel('Actual'); ylabel('Recovered');

subplot(3, 2, 6);

scatter(parsPlot.gamma(parsPlot.epsilon == 0.5 & parsPlot.beta > 5), parsPlot.gammaRecovered(parsPlot.epsilon == 0.5 & parsPlot.beta > 5));

hold on; plot(xlim, ylim, '--k'); hold off;

title('Gamma'); lsline; xlabel('Actual'); ylabel('Recovered');

There does seem to be a positive correlation between these parameters, but it's not particularly reliable. Beta Epsilon, and Gamma are the strongest, Alpha and Delta are pretty poor and Rho seems to be pretty random

2.1 Recovering Free Parameters

First let’s get the trial data from participants

trialData = readtable("C:/Users/DELL/Downloads/Data/Data/HPP_fMRI_beh_data_for_lmm.csv")
trialData = 9104×51 table 
 subjecttrial_numrunpartnerApartnerBinitial_Ainitial_BPie_initial_highPie_initial_lowdiff_initialtransfer_size_1fairness_1A_final_1B_final_1Pie_Alter1_ini_highPie_Alter1_ini_lowdiff_final_1reverse_1transfer_size_2fairness_2A_final_2B_final_2Pie_Alter2_ini_highPie_Alter2_ini_lowEqual_alter_pie_ini_highEqual_alter_pie_ini_lowUnqual_alter_pie_ini_highUnequal_alter_pie_ini_lowdiff_final_2reverse_2transfer_difffairness_difftrail_typeorder_optionorder_orientationHPP_keyreactionIf_RevIf_Rev_1Revsize1Revsize2TransferSize1_aTransferSize2_aTransferSize1_bTransferSize2_bMahMalMbhMblTrans1_DiffTrans2_Diff
110111464315215213510107107301214314310714311048121100000501010714340
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410141424715315312510108108201214414410814410048121100000501010814440
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9101912045163163131215415411011451451451415491102311111109106.50004.50001545145.50004.5000
101011011653152152136129898102413413498134904812110000060209813440
11101111204551515510368121284051010101010101012801244122011003050128101020
1210112132211161611551061111650714899898116102411211000050701169820
13101131222351616511610111010111124714147101114770463221011105.50000.50002010111473.50000.5000
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19101191622121515213810107710312441313471013490663221011306.50001.5000207101344.50001.5000
2010120182131515312612999901961266129961261-3652210110660639961203
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33101331543315115114510106106401214214210614212048121210000501010614240
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3510135167351616511510101111101024714147111014770361221000005020111014730
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4710147130251531531248117117405101081081081172012111200000405011710810
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49101491525711616115510611116508149889891161134312201100507.50000.5000116892.50000.5000
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521015211457162162148128108102171499999981001-1251120112071708109901
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57101572341731616313810118811312451414581114590663221011306.50001.5000208111454.50001.5000
581015823035153153123612612660481171171171264012111110000304012611710
591015926672151521312314143110365121251251437024112200000103014312520
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621016227313161631336613136708101188118111363154312201100306.50001.50001368113.50001.5000
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91101912641915115114121421421201243133133131421011123111111010107514231365
9210192246471631631361210910910241451451091459048121210000602010914540
931019326465151151147148888011164124128841281-4852110110870748841204
94101942767713213211610787811121231237812390583211011105.50000.500010781234.50000.5000
9510195224252161621471499990151071111799117402442221110070509911720
9610196264633161631361291010910366131361091367036122100000603010913630
9710197222214161641236713137607101199119111372144312201100306113791131
98101982202521616214910117711412441414471114410076322101140722071114452
991019928693161631351081111830245141451181459036122100000502011814530
100101100234351151511448511115601221414211514212036122100000401011514230
⋮

So we need participants' ID numbers, a0-b2, and their choices in the format 1 or 2. In the above data frame, those are columns 1, 6, 7, 13, 14, 21, 22, and 26 so let's extract those. Also, they only analyzed the rank-reverse condition (perhaps this is why we couldn't reliably recover parameters) so that is "trail_type" 3.

trialData = trialData(trialData.trail_type == 3, [1, 6, 7, 13, 14, 21, 22, 36]);
trialData.Properties.VariableNames = {'SubjectID', 'a0', 'b0', 'a1', 'b1', 'a2', 'b2', 'Chose1'};
trialData.Chose1 = trialData.Chose1 - 1;
trialData.Prob1 = zeros(height(trialData), 1)
trialData = 3704×9 table 
 SubjectIDa0b0a1b1a2b2Chose1Prob1
110116514781310
210116151215200
310116315451400
4101516111071400
51011528913400
610151671412910
710111512421400
810121510741300
910115381014400
1010131510851300
1110115161014200
1210131511741410
1310115311781000
1410114213341200
1510115271014300
1610131612741500
1710115211671000
1810121651311700
1910121641412610
201011166119810
2110131651412710
2210116213561200
2310115314451310
2410131611851400
2510131541412610
2610131661311810
2710116391014500
2810115212561110
291011159741200
301011517913310
3110121613531510
3210121531412500
3310116515671400
3410115312671100
3510116271115310
3610151661513800
3710115114231300
381011327812300
3910141671311910
4010121611741400
4110116211781010
4210111511521400
4310116381115410
4410131610961300
451012149741200
4610121410631310
4710116215341410
4810131641513610
4910151612961510
5010131551311700
51101165101115610
5210116171014310
5310121541311600
5410111613421500
5510116281014410
5610121612631510
5710115313561200
5810121661210810
5910116212671110
6010116513891210
6110115214341300
6210116161115210
631011427913300
6410111531311510
6510115213451200
6610121511631410
6710216514781300
6810216151215200
6910216315451410
70102516111071400
711021528913400
7210251671412900
7310211512421410
7410221510741300
7510215381014410
7610231510851310
7710215161014210
7810231511741410
7910215311781000
8010214213341200
8110215271014310
8210231612741510
8310215211671000
8410221651311700
8510221641412600
861021166119800
8710231651412710
8810216213561200
8910215314451300
9010231611851410
9110231541412600
9210231661311800
9310216391014510
9410215212561100
951021159741210
961021517913310
9710221613531510
9810221531412500
9910216515671400
10010215312671100
⋮

	SubjectID	a0	b0	a1	b1	a2	b2	Chose1	Prob1
1	101	16	5	14	7	8	13	1	0
2	101	16	1	5	12	15	2	0	0
3	101	16	3	15	4	5	14	0	0
4	101	5	16	11	10	7	14	0	0
5	101	15	2	8	9	13	4	0	0
6	101	5	16	7	14	12	9	1	0
7	101	1	15	12	4	2	14	0	0
8	101	2	15	10	7	4	13	0	0
9	101	15	3	8	10	14	4	0	0
10	101	3	15	10	8	5	13	0	0
11	101	15	1	6	10	14	2	0	0
12	101	3	15	11	7	4	14	1	0
13	101	15	3	11	7	8	10	0	0
14	101	14	2	13	3	4	12	0	0
15	101	15	2	7	10	14	3	0	0
16	101	3	16	12	7	4	15	0	0
17	101	15	2	11	6	7	10	0	0
18	101	2	16	5	13	11	7	0	0
19	101	2	16	4	14	12	6	1	0
20	101	1	16	6	11	9	8	1	0
21	101	3	16	5	14	12	7	1	0
22	101	16	2	13	5	6	12	0	0
23	101	15	3	14	4	5	13	1	0
24	101	3	16	11	8	5	14	0	0
25	101	3	15	4	14	12	6	1	0
26	101	3	16	6	13	11	8	1	0
27	101	16	3	9	10	14	5	0	0
28	101	15	2	12	5	6	11	1	0
29	101	1	15	9	7	4	12	0	0
30	101	15	1	7	9	13	3	1	0
31	101	2	16	13	5	3	15	1	0
32	101	2	15	3	14	12	5	0	0
33	101	16	5	15	6	7	14	0	0
34	101	15	3	12	6	7	11	0	0
35	101	16	2	7	11	15	3	1	0
36	101	5	16	6	15	13	8	0	0
37	101	15	1	14	2	3	13	0	0
38	101	13	2	7	8	12	3	0	0
39	101	4	16	7	13	11	9	1	0
40	101	2	16	11	7	4	14	0	0
41	101	16	2	11	7	8	10	1	0
42	101	1	15	11	5	2	14	0	0
43	101	16	3	8	11	15	4	1	0
44	101	3	16	10	9	6	13	0	0
45	101	2	14	9	7	4	12	0	0
46	101	2	14	10	6	3	13	1	0
47	101	16	2	15	3	4	14	1	0
48	101	3	16	4	15	13	6	1	0
49	101	5	16	12	9	6	15	1	0
50	101	3	15	5	13	11	7	0	0
51	101	16	5	10	11	15	6	1	0
52	101	16	1	7	10	14	3	1	0
53	101	2	15	4	13	11	6	0	0
54	101	1	16	13	4	2	15	0	0
55	101	16	2	8	10	14	4	1	0
56	101	2	16	12	6	3	15	1	0
57	101	15	3	13	5	6	12	0	0
58	101	2	16	6	12	10	8	1	0
59	101	16	2	12	6	7	11	1	0
60	101	16	5	13	8	9	12	1	0
61	101	15	2	14	3	4	13	0	0
62	101	16	1	6	11	15	2	1	0
63	101	14	2	7	9	13	3	0	0
64	101	1	15	3	13	11	5	1	0
65	101	15	2	13	4	5	12	0	0
66	101	2	15	11	6	3	14	1	0
67	102	16	5	14	7	8	13	0	0
68	102	16	1	5	12	15	2	0	0
69	102	16	3	15	4	5	14	1	0
70	102	5	16	11	10	7	14	0	0
71	102	15	2	8	9	13	4	0	0
72	102	5	16	7	14	12	9	0	0
73	102	1	15	12	4	2	14	1	0
74	102	2	15	10	7	4	13	0	0
75	102	15	3	8	10	14	4	1	0
76	102	3	15	10	8	5	13	1	0
77	102	15	1	6	10	14	2	1	0
78	102	3	15	11	7	4	14	1	0
79	102	15	3	11	7	8	10	0	0
80	102	14	2	13	3	4	12	0	0
81	102	15	2	7	10	14	3	1	0
82	102	3	16	12	7	4	15	1	0
83	102	15	2	11	6	7	10	0	0
84	102	2	16	5	13	11	7	0	0
85	102	2	16	4	14	12	6	0	0
86	102	1	16	6	11	9	8	0	0
87	102	3	16	5	14	12	7	1	0
88	102	16	2	13	5	6	12	0	0
89	102	15	3	14	4	5	13	0	0
90	102	3	16	11	8	5	14	1	0
91	102	3	15	4	14	12	6	0	0
92	102	3	16	6	13	11	8	0	0
93	102	16	3	9	10	14	5	1	0
94	102	15	2	12	5	6	11	0	0
95	102	1	15	9	7	4	12	1	0
96	102	15	1	7	9	13	3	1	0
97	102	2	16	13	5	3	15	1	0
98	102	2	15	3	14	12	5	0	0
99	102	16	5	15	6	7	14	0	0
100	102	15	3	12	6	7	11	0	0
⋮

Now we can define some necessary variables and functions

included_subjects = unique(trialData.SubjectID);
 
subjectData = table();
 
grab_data = @(subject) trialData(trialData.SubjectID == subject, 2:end);
 
function updated = addPredictions(trialData, subject, predictions)
trialData.Prob1(trialData.SubjectID == subject) = predictions;
updated = trialData;
end

Which allows us to Recover Free Parameters and Define Predicted Decisions

for i = 1:length(included_subjects)
    df = grab_data(included_subjects(i));
    result = optimize(@obj_function, initial_params, lower_bounds, upper_bounds, df);
 
    df.Prob1 = generatePredictions(result, df);
    
    model_SS = sum((df.Chose1 - df.Prob1).^2);
    model_NLL = -2 * sum(df.Chose1 .* log(df.Prob1) + (1 - df.Chose1) .* log(1 - df.Prob1));
    
    subjectData(i, :) = {included_subjects(i), result(1), result(2), result(3), result(4), result(5), result(6), model_SS, model_NLL};
    
    trialData = addPredictions(trialData, included_subjects(i), df.Prob1);
end
subjectData.Properties.VariableNames = {'subjectID', 'Alpha', 'Delta', 'Rho', 'Beta', 'Epsilon', 'Gamma', 'SS', 'Deviance'}
subjectData = 57×9 table 
 subjectIDAlphaDeltaRhoBetaEpsilonGammaSSDeviance
11010.41361.43231.05774.20550.4394-0.051716.121289.9689
21021.27810.08971.30415.71550.1256-0.34477.352248.5749
31031.58950.10320.62477.01360.0157-0.49810.96888.9042
41041.67090.07400.52217.76460.01520.49800.96978.9665
51050.51151.39431.29664.49850.0313-0.49901.875014.9667
61061.69460.07680.54458.05680.1529-0.19358.233954.0180
71070.47451.51791.10494.36730.2273-0.100011.454569.9661
81081.98241.35078.0938e-079.91940.4165-0.191915.510387.4497
91090.43101.45411.05304.42650.3437-0.272712.187571.9739
101101.0132e-060.19292.5954e-064.14370.0577-0.50003.438824.0005
111111.58580.10390.61356.96254.2555e-05-0.00621.4398e-070.0060
121121.63561.67262.1653e-069.44040.2840-0.129613.925480.4270
131131.58580.10390.61356.96254.2555e-05-0.00621.4398e-070.0060
141141.05760.03581.47006.37090.1822-0.33169.659560.1556
151151.57440.10180.62226.86100.1364-0.49986.545538.6772
161161.59720.10260.66776.98380.0909-0.49974.909131.2975
171171.42591.94371.95089.87620.2435-0.060012.009772.2680
181181.68350.07950.47317.87190.1979-0.34219.208356.9726
191191.58730.10380.62697.04310.0152-0.49800.96978.9667
201201.70040.07560.52707.74860.3987-0.108115.117485.4669
211210.72720.06670.13014.11870.3379-0.239912.907375.6418
221221.58570.10390.61346.96324.2555e-05-0.00631.4398e-070.0060
231240.64940.09210.11903.75900.28600.022613.819580.4151
241251.37940.12550.00202.24760.02560.49881.874015.0043
251261.17920.78900.90544.17530.5000-0.250012.000071.9789
261270.66931.20871.09363.99100.5000-0.075816.121289.9744
271281.58900.10500.61627.04370.0455-0.49932.727320.1103
281290.97420.76862.000010.00000.2698-0.283713.441077.5883
291301.59520.09950.57866.53400.0157-0.49810.96888.9041
301311.64930.07400.40947.76470.0909-0.16675.393939.4657
311320.41691.42411.09764.41550.1212-0.37506.484843.0681
321330.70841.20334.5178e-060.23280.2110-0.393713.822580.5423
331340.72601.23391.05534.00470.5000-0.084615.784688.2386
341350.74420.53422.000010.00000.3065-0.243314.014480.4176
351361.66650.07140.54787.74690.1061-0.35715.878840.2556
361380.44291.46701.23194.06190.3571-0.200012.214370.6853
371401.50161.54221.99979.99870.3678-0.005815.137785.6076
381411.99881.71301.99979.99900.4157-0.101115.333785.6393
391420.77241.17491.05314.06790.5000-0.060616.257690.5233
401430.58521.37121.11574.00800.5000-0.115415.384686.6162
411441.58520.10320.62376.97780.0162-0.49810.96778.8396
421450.75591.19490.97634.11780.5000-0.129014.467781.7741
431460.77591.17891.04284.03850.5000-0.136415.272786.5236
441470.62601.25151.18704.50354.8527e-050.02931.6811e-070.0066
451481.60860.10350.64506.88640.0909-0.49974.909131.2975
461491.99970.36355.5472e-049.99862.8567e-05-0.31390.50132.8825
471500.75850.58012.00009.99990.2887-0.318912.439472.0456
481511.61120.06910.45438.10680.2214-0.17749.728659.6184
491520.91365.8932e-071.56627.08970.1727-0.114010.359664.9554
501530.59631.37841.02565.95770.40910.055615.818288.7364
511541.68220.08100.48127.80660.0303-1.1914e-071.939417.9248
521552.00000.36273.8484e-059.99990.04690.17753.347226.6615
531560.62921.25471.0216e-051.65160.0310-0.38742.766421.6691
541570.52122.1390e-061.28951.39130.1660-0.500013.289777.3106
551581.56121.99931.99939.99820.4772-0.066416.223990.3865
561600.38471.41311.06444.19580.15340.09268.409155.4065
571610.43221.37000.87574.67770.4129-0.197214.060680.9767

	subjectID	Alpha	Delta	Rho	Beta	Epsilon	Gamma	SS	Deviance
1	101	0.4136	1.4323	1.0577	4.2055	0.4394	-0.0517	16.1212	89.9689
2	102	1.2781	0.0897	1.3041	5.7155	0.1256	-0.3447	7.3522	48.5749
3	103	1.5895	0.1032	0.6247	7.0136	0.0157	-0.4981	0.9688	8.9042
4	104	1.6709	0.0740	0.5221	7.7646	0.0152	0.4980	0.9697	8.9665
5	105	0.5115	1.3943	1.2966	4.4985	0.0313	-0.4990	1.8750	14.9667
6	106	1.6946	0.0768	0.5445	8.0568	0.1529	-0.1935	8.2339	54.0180
7	107	0.4745	1.5179	1.1049	4.3673	0.2273	-0.1000	11.4545	69.9661
8	108	1.9824	1.3507	8.0938e-07	9.9194	0.4165	-0.1919	15.5103	87.4497
9	109	0.4310	1.4541	1.0530	4.4265	0.3437	-0.2727	12.1875	71.9739
10	110	1.0132e-06	0.1929	2.5954e-06	4.1437	0.0577	-0.5000	3.4388	24.0005
11	111	1.5858	0.1039	0.6135	6.9625	4.2555e-05	-0.0062	1.4398e-07	0.0060
12	112	1.6356	1.6726	2.1653e-06	9.4404	0.2840	-0.1296	13.9254	80.4270
13	113	1.5858	0.1039	0.6135	6.9625	4.2555e-05	-0.0062	1.4398e-07	0.0060
14	114	1.0576	0.0358	1.4700	6.3709	0.1822	-0.3316	9.6595	60.1556
15	115	1.5744	0.1018	0.6222	6.8610	0.1364	-0.4998	6.5455	38.6772
16	116	1.5972	0.1026	0.6677	6.9838	0.0909	-0.4997	4.9091	31.2975
17	117	1.4259	1.9437	1.9508	9.8762	0.2435	-0.0600	12.0097	72.2680
18	118	1.6835	0.0795	0.4731	7.8719	0.1979	-0.3421	9.2083	56.9726
19	119	1.5873	0.1038	0.6269	7.0431	0.0152	-0.4980	0.9697	8.9667
20	120	1.7004	0.0756	0.5270	7.7486	0.3987	-0.1081	15.1174	85.4669
21	121	0.7272	0.0667	0.1301	4.1187	0.3379	-0.2399	12.9073	75.6418
22	122	1.5857	0.1039	0.6134	6.9632	4.2555e-05	-0.0063	1.4398e-07	0.0060
23	124	0.6494	0.0921	0.1190	3.7590	0.2860	0.0226	13.8195	80.4151
24	125	1.3794	0.1255	0.0020	2.2476	0.0256	0.4988	1.8740	15.0043
25	126	1.1792	0.7890	0.9054	4.1753	0.5000	-0.2500	12.0000	71.9789
26	127	0.6693	1.2087	1.0936	3.9910	0.5000	-0.0758	16.1212	89.9744
27	128	1.5890	0.1050	0.6162	7.0437	0.0455	-0.4993	2.7273	20.1103
28	129	0.9742	0.7686	2.0000	10.0000	0.2698	-0.2837	13.4410	77.5883
29	130	1.5952	0.0995	0.5786	6.5340	0.0157	-0.4981	0.9688	8.9041
30	131	1.6493	0.0740	0.4094	7.7647	0.0909	-0.1667	5.3939	39.4657
31	132	0.4169	1.4241	1.0976	4.4155	0.1212	-0.3750	6.4848	43.0681
32	133	0.7084	1.2033	4.5178e-06	0.2328	0.2110	-0.3937	13.8225	80.5423
33	134	0.7260	1.2339	1.0553	4.0047	0.5000	-0.0846	15.7846	88.2386
34	135	0.7442	0.5342	2.0000	10.0000	0.3065	-0.2433	14.0144	80.4176
35	136	1.6665	0.0714	0.5478	7.7469	0.1061	-0.3571	5.8788	40.2556
36	138	0.4429	1.4670	1.2319	4.0619	0.3571	-0.2000	12.2143	70.6853
37	140	1.5016	1.5422	1.9997	9.9987	0.3678	-0.0058	15.1377	85.6076
38	141	1.9988	1.7130	1.9997	9.9990	0.4157	-0.1011	15.3337	85.6393
39	142	0.7724	1.1749	1.0531	4.0679	0.5000	-0.0606	16.2576	90.5233
40	143	0.5852	1.3712	1.1157	4.0080	0.5000	-0.1154	15.3846	86.6162
41	144	1.5852	0.1032	0.6237	6.9778	0.0162	-0.4981	0.9677	8.8396
42	145	0.7559	1.1949	0.9763	4.1178	0.5000	-0.1290	14.4677	81.7741
43	146	0.7759	1.1789	1.0428	4.0385	0.5000	-0.1364	15.2727	86.5236
44	147	0.6260	1.2515	1.1870	4.5035	4.8527e-05	0.0293	1.6811e-07	0.0066
45	148	1.6086	0.1035	0.6450	6.8864	0.0909	-0.4997	4.9091	31.2975
46	149	1.9997	0.3635	5.5472e-04	9.9986	2.8567e-05	-0.3139	0.5013	2.8825
47	150	0.7585	0.5801	2.0000	9.9999	0.2887	-0.3189	12.4394	72.0456
48	151	1.6112	0.0691	0.4543	8.1068	0.2214	-0.1774	9.7286	59.6184
49	152	0.9136	5.8932e-07	1.5662	7.0897	0.1727	-0.1140	10.3596	64.9554
50	153	0.5963	1.3784	1.0256	5.9577	0.4091	0.0556	15.8182	88.7364
51	154	1.6822	0.0810	0.4812	7.8066	0.0303	-1.1914e-07	1.9394	17.9248
52	155	2.0000	0.3627	3.8484e-05	9.9999	0.0469	0.1775	3.3472	26.6615
53	156	0.6292	1.2547	1.0216e-05	1.6516	0.0310	-0.3874	2.7664	21.6691
54	157	0.5212	2.1390e-06	1.2895	1.3913	0.1660	-0.5000	13.2897	77.3106
55	158	1.5612	1.9993	1.9993	9.9982	0.4772	-0.0664	16.2239	90.3865
56	160	0.3847	1.4131	1.0644	4.1958	0.1534	0.0926	8.4091	55.4065
57	161	0.4322	1.3700	0.8757	4.6777	0.4129	-0.1972	14.0606	80.9767

2.2 Compute Model Fit Index

We will calculate BIC as the model fit index because we are attempting to model the probabilistic nature of the data generation process.

subjectData.BIC = subjectData.Deviance + log(65) * 6;

2.3 Identify the Best Model

We need to define new objective functions for each model. Since each model uses the same utility function, but holds some variables constant (at 0), we only really need to modify the number of parameter inputs and set the constant values to 0. We can also use a list of indices so that the we can use the same function to generate predictions down the line.

function f = of_alphaOnly(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new([1, 4:6]) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
function f = of_deltaOnly(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new([2, 4:6]) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
function f = of_rhoOnly(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new(3:6) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
function f = of_ad(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new([1:2, 4:6]) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
function f = of_ar(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new([1, 3:6]) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
function f = of_dr(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new(2:6) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
function f = of_noEpsilon(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new(1:4) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
function f = of_noGamma(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new(1:5) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
function f = of_GammaOnly(params, df, optimMethod)
    if nargin < 3
        optimMethod = 'MLE';
    end
    params_new = zeros(1, 6);
    params_new(5) = 0.5;
    params_new(6) = params;
    Prob1 = generatePredictions(params_new, df);
    Chose1 = df.Chose1;
    if strcmp(optimMethod, 'OLS')
        f = sum((Chose1 - Prob1).^2);
    elseif strcmp(optimMethod, 'MLE')
        f = -sum(Chose1 .* log(Prob1) + (1 - Chose1) .* log(1 - Prob1));
    end
end
 
ofs = {@of_alphaOnly, @of_deltaOnly, @of_rhoOnly, @of_ad, @of_ar, @of_dr, @of_noEpsilon, @of_noGamma, @of_GammaOnly};
idxs = {[1, 4:6], [2, 4:6], 3:6, [1:2, 4:6], [1, 3:6], 2:6, 1:4, 1:5, 6};

And now we can preallocate the predictions for each model and the new data frame

altSubjectData = table();
 
altTrialData = trialData;
altTrialData.Prob1 = []; % Remove the 9th column
altTrialData.alphaOnly_Prob1 = zeros(height(altTrialData), 1);
altTrialData.deltaOnly_Prob1 = zeros(height(altTrialData), 1);
altTrialData.rhoOnly_Prob1 = zeros(height(altTrialData), 1);
altTrialData.ad_Prob1 = zeros(height(altTrialData), 1);
altTrialData.ar_Prob1 = zeros(height(altTrialData), 1);
altTrialData.dr_Prob1 = zeros(height(altTrialData), 1);
altTrialData.noEpsilon_Prob1 = zeros(height(altTrialData), 1);
altTrialData.noGamma_Prob1 = zeros(height(altTrialData), 1);
altTrialData.gammaOnly_Prob1 = zeros(height(altTrialData), 1);

Now let's Recover Free Parameters and Generate Predictions for this Model

for i = 1:length(included_subjects)
    df = grab_data(included_subjects(i));
    numParams = sum(cellfun(@length, idxs)) + 2 * 9;
    outputs = zeros(1, numParams);
    j = 0;
 
    for k = 1:length(idxs)
        idx = idxs{k};
        initials = initial_params(idx);
        uppers = upper_bounds(idx);
        lowers = lower_bounds(idx);
        of = ofs{k};
 
        result = optimize(of, initials, lowers, uppers, df);
 
        pars = zeros(1, 6);
        pars(idx) = result;
        df.Prob1 = generatePredictions(pars, df);
 
        model_SS = sum((df.Chose1 - df.Prob1).^2);
        model_NLL = -2 * sum(df.Chose1 .* log(df.Prob1) + (1 - df.Chose1) .* log(1 - df.Prob1));
        outputs((j+1):(j+2+length(result))) = [result, model_SS, model_NLL];
        j = j + 2 + length(result);
 
        altTrialData{altTrialData.SubjectID == included_subjects(i), (8+k)} = df.Prob1;
    end
    altSubjectData(i, 1:56) = array2table([included_subjects(i), outputs]);
end
 
altSubjectData.Properties.VariableNames = {'SubjectID', 'Alpha_M1', 'Beta_M1', 'Epsilon_M1', 'Gamma_M1', 'SS_M1', 'Deviance_M1', ...
                                           'Delta_M2', 'Beta_M2', 'Epsilon_M2', 'Gamma_M2', 'SS_M2', 'Deviance_M2', ...
                                           'Rho_M3', 'Beta_M3', 'Epsilon_M3', 'Gamma_M3', 'SS_M3', 'Deviance_M3', ...
                                           'Alpha_M4', 'Delta_M4', 'Beta_M4', 'Epsilon_M4', 'Gamma_M4', 'SS_M4', 'Deviance_M4', ...
                                           'Alpha_M5', 'Rho_M5', 'Beta_M5', 'Epsilon_M5', 'Gamma_M5', 'SS_M5', 'Deviance_M5', ...
                                           'Delta_M6', 'Rho_M6', 'Beta_M6', 'Epsilon_M6', 'Gamma_M6', 'SS_M6', 'Deviance_M6', ...
                                           'Alpha_M7', 'Delta_M7', 'Rho_M7', 'Beta_M7', 'SS_M7', 'Deviance_M7', ...
                                           'Alpha_M8', 'Delta_M8', 'Rho_M8', 'Beta_M8', 'Epsilon_M8', 'SS_M8', 'Deviance_M8', ...
                                           'Gamma_M9', 'SS_M9', 'Deviance_M9'};
 
altSubjectData(:, 2:end) = varfun(@double, altSubjectData(:, 2:end))
altSubjectData = 57×56 table 
 SubjectIDAlpha_M1Beta_M1Epsilon_M1Gamma_M1SS_M1Deviance_M1Delta_M2Beta_M2Epsilon_M2Gamma_M2SS_M2Deviance_M2Rho_M3Beta_M3Epsilon_M3Gamma_M3SS_M3Deviance_M3Alpha_M4Delta_M4Beta_M4Epsilon_M4Gamma_M4SS_M4Deviance_M4Alpha_M5Rho_M5Beta_M5Epsilon_M5Gamma_M5SS_M5Deviance_M5Delta_M6Rho_M6Beta_M6Epsilon_M6Gamma_M6SS_M6Deviance_M6Alpha_M7Delta_M7Rho_M7Beta_M7SS_M7Deviance_M7Alpha_M8Delta_M8Rho_M8Beta_M8Epsilon_M8SS_M8Deviance_M8Gamma_M9SS_M9Deviance_M9
11010.99964.02040.5000-0.045516.363690.94921.17554.07660.4394-0.051716.121289.96890.92393.96470.4362-0.052116.121289.96890.23451.45384.34290.4394-0.051716.121289.96890.98761.00594.01660.5000-0.045516.363690.94921.00001.00004.01840.4394-0.051716.121289.96895.2475e-072.4427e-071.34790.180716.257690.52330.44121.45971.06914.11160.439416.257690.5233-0.045516.500091.4954
21021.62954.36510.1364-0.27787.393949.19541.00004.03810.5000-0.075816.121289.97440.19163.58480.5000-0.075816.121289.97441.15130.16513.12180.1256-0.34447.352348.57291.47840.41835.19710.1364-0.27787.393949.19541.00001.00004.03810.5000-0.075816.121289.97441.38390.18704.5093e-042.11358.8712284.20211.60390.08580.39617.62610.13647.772752.5765-0.075816.500091.4954
31031.36984.17020.0157-0.49790.96888.90470.99994.02690.5000-0.023116.215489.97060.10243.41580.5000-0.023116.215489.97061.48620.13286.65320.0157-0.49810.96888.90411.33810.81346.31710.0157-0.49780.96888.90461.00001.00004.02690.5000-0.023116.215489.97061.54800.07280.47745.00981.000046.05311.67140.07920.53777.74960.01540.984610.3333-0.023116.250090.1091
41040.51142.77833.2654e-06-0.13730.94477.85350.99994.01630.50000.015216.484991.43480.02371.46940.49740.015216.487791.44641.59140.09897.50940.01520.49800.96978.96650.87761.05864.06825.0665e-050.05210.94447.73051.00001.00004.01630.50000.015216.484991.43481.16630.00630.91742.09080.94867.82161.64180.06910.56217.50300.01520.984810.36410.015216.500091.4954
51050.99054.02250.5000-0.031215.937588.47270.18204.08230.0208-0.50001.800614.80221.50687.55550.0313-0.49911.875014.96714.0444e-061.03250.71970.0208-0.50001.800614.80220.02781.82049.01390.0313-0.49901.875014.96670.20373.2526e-063.64860.0208-0.50001.800614.80221.8653e-060.74611.0950e-050.85981.841418.83930.47951.38701.10754.41810.03131.937517.7998-0.03121688.7228
61061.32126.60750.1529-0.19358.233954.01801.00004.02600.5000-0.053816.061589.35388.4391e-043.34990.4994-0.053916.061689.35401.61090.09867.58990.1529-0.19358.233954.01801.61530.60998.01100.1529-0.19358.233954.01801.00001.00004.02600.5000-0.053816.061589.35381.65020.08550.59054.777110.0000473.74531.70220.07400.51348.00400.15388.461555.8120-0.053816.250090.1091
71070.99284.02490.5000-0.045516.363690.94921.37874.24270.2273-0.100011.454569.96611.01754.03780.2182-0.104211.454569.96610.28271.61904.05340.2273-0.100011.454569.96610.01711.72088.26530.2273-0.100011.454569.96611.15381.05134.15270.2273-0.100011.454569.96615.0894e-083.1019e-070.31823.845911.590970.74670.37091.35231.24304.26900.227311.590970.7467-0.045516.500091.4954
81081.00114.06230.5000-0.106115.757688.50310.01173.63710.1054-0.499715.524387.55550.95294.76010.4690-0.113115.697088.24900.79530.65731.39500.3935-0.204015.495787.46191.01390.99284.01360.5000-0.106115.757688.50311.00001.00004.02470.4697-0.112915.697088.24902.0441e-061.33451.0884e-050.025716.257390.51921.41880.29560.82114.15540.500016.500091.4954-0.106116.500091.4954
91090.98494.06620.5000-0.187513.750079.49911.16724.07600.3437-0.272712.187571.97390.99574.98790.3416-0.274512.187571.97390.33951.61474.46530.3437-0.272712.187571.97390.23871.99999.99960.3209-0.327712.105371.35911.00001.00004.00470.3437-0.272712.187571.97396.3150e-090.00950.05755.277614.404782.22800.28921.31991.05114.18610.343714.437582.3669-0.18751688.7228
101100.98974.01580.5000-0.064515.241984.91510.26423.02560.0577-0.50003.438824.00051.49757.48730.0645-0.49953.483923.84605.4358e-061.03500.77230.0577-0.50003.438824.00050.02941.81678.98770.0645-0.49953.483923.84571.01891.3708e-050.78450.0577-0.50003.438824.00054.1573e-070.55173.1257e-050.91533.732532.30450.55951.31541.07185.88120.06453.741929.6629-0.064515.500085.9503
111111.11495.57435.4500e-05-0.00342.0413e-070.00730.99993.98210.50001.4217e-0816.500091.49540.03433.44610.4991-7.3164e-0616.503791.51021.49070.13526.67394.2872e-05-0.01111.3984e-070.00601.33170.81956.28154.7994e-05-0.00871.6948e-070.00671.00001.00003.99000.50003.4442e-0916.500091.49541.58100.09890.62686.40282.4791e-088.0190e-041.58370.10300.62346.97334.2490e-051.4420e-070.0060016.500091.4954
121120.99104.01480.50000.023116.215489.97060.08971.24210.02690.499814.181181.51380.94073.99620.33030.038714.511483.01131.62651.66339.48700.2840-0.129613.925480.42700.25301.99999.99960.2865-0.045314.060681.06580.20321.3894e-070.54820.02690.500014.181181.51380.71541.15870.16620.154013.981480.73361.26111.33434.5506e-066.46420.319714.196481.70870.023116.250090.1091
131131.11495.57435.4500e-05-0.00342.0413e-070.00730.99993.98210.50001.4217e-0816.500091.49540.03433.44610.4991-7.3164e-0616.503791.51021.49070.13526.67394.2872e-05-0.01111.3984e-070.00601.33170.81956.28154.7994e-05-0.00871.6948e-070.00671.00001.00003.99000.50003.4442e-0916.500091.49541.58100.09890.62686.40282.4791e-088.0190e-041.58370.10300.62346.97334.2490e-051.4420e-070.0060016.500091.4954
141140.33433.83200.1835-0.32449.681260.39351.00004.08750.5000-0.106115.757688.50311.8101e-072.61610.2888-0.183615.757688.50311.07940.12151.95240.1824-0.33059.660360.16630.91230.56662.02550.1841-0.32259.680760.39041.00001.00004.08750.5000-0.106115.757688.50311.33775.1408e-073.8893e-060.229712.450479.73451.70540.07040.57667.62460.197010.439465.4948-0.106116.500091.4954
151151.11715.58510.1364-0.49976.545538.67751.00004.11590.5000-0.136415.272786.52363.6313e-075.49070.4574-0.149115.272786.52361.47490.13446.64210.1364-0.49986.545538.67711.30100.60525.66580.1364-0.49986.545538.67761.00001.00004.11600.5000-0.136415.272786.52361.58820.09860.64506.51179.0000414.46601.68600.07000.54648.00860.13647.772752.5765-0.136416.500091.4954
161161.12265.61290.0909-0.49964.909131.29781.00004.08180.5000-0.090915.954589.30145.8531e-043.07480.4961-0.091615.954789.30221.47510.13276.64710.0909-0.49974.909131.29741.29900.66426.01690.0909-0.49974.909131.29741.00001.00004.08190.5000-0.090915.954589.30141.54600.06110.36074.41326.0000276.31201.70560.07620.53787.96710.09095.454540.2120-0.090916.500091.4954
171170.99324.01520.5000-0.023116.215489.97061.36714.23840.2457-0.054912.014272.29340.98254.01500.2357-0.057312.014272.29340.24641.34795.05950.2457-0.054912.014272.29340.25282.000010.00000.1623-0.339810.927665.69191.09331.03344.10260.2457-0.054912.014272.29344.5155e-074.2333e-071.35260.827412.061572.54910.41631.46021.04364.15100.246212.061572.5491-0.023116.250090.1091
181181.68754.25750.1979-0.34219.208356.97261.00004.09720.5000-0.130815.138585.61091.1343e-073.17830.2174-0.300815.138585.61091.59120.09377.42900.1979-0.34219.208356.97261.49230.75126.95520.1979-0.34219.208356.97261.00001.00004.09720.5000-0.130815.138585.61091.35124.1967e-071.7084e-050.970212.9343233.44831.69510.06890.48637.93530.200010.400065.0523-0.130816.250090.1091
191191.34724.16920.0152-0.49770.96978.96740.99994.01630.5000-0.015216.484991.43480.02301.48730.4974-0.015216.487791.44641.49340.13516.68850.0152-0.49800.96978.96671.60100.67624.20360.0152-0.49790.96978.96741.00001.00004.01630.5000-0.015216.484991.43481.56660.08830.48645.58501.000046.05261.66050.08130.50877.78680.01520.984810.3641-0.015216.500091.4954
201201.77794.23650.3987-0.108115.117485.46691.00004.08990.5000-0.084615.784688.23860.04173.57510.4998-0.084715.784888.23941.25020.76773.86690.5000-0.084615.784688.23861.75630.25684.85300.3987-0.108115.117485.46691.00001.00004.08990.5000-0.084615.784688.23861.04340.52970.55343.3489e-0816.250090.10911.23890.67920.98153.79330.500016.250090.1091-0.084616.250090.1091
211211.57604.20960.3409-0.233312.909175.65461.00004.05930.5000-0.161514.553883.20135.2627e-083.55980.2926-0.276014.553883.20131.12210.11332.39050.3376-0.240512.907375.64171.53350.81036.91860.3409-0.233312.909175.65461.00001.00004.05940.5000-0.161514.553883.20130.02788.1461e-096.3605e-083.102815.478187.02110.34181.61161.04744.02460.500016.250090.1091-0.161516.250090.1091
221221.11495.57425.4500e-05-0.00342.0413e-070.00730.99993.98210.50001.3140e-0816.500091.49540.03423.44540.4990-7.7564e-0616.503791.51021.49060.13526.67384.2872e-05-0.01111.3984e-070.00601.33200.81966.27994.7989e-05-0.00841.6945e-070.00671.00001.00003.98910.50009.3726e-0916.500091.49551.58100.09890.62686.40282.4791e-088.0190e-041.58470.10250.63066.97474.2481e-051.4425e-070.0060016.500091.4954
231241.24974.08800.30300.050013.878880.68281.00004.03400.50000.030316.439491.25299.8594e-043.45090.49910.030416.439491.25301.07220.16722.04520.28550.022213.819480.41511.40590.79614.09250.30300.050013.878880.68281.00001.00004.03410.50000.030316.439491.25291.36895.2492e-064.0752e-060.120713.953280.94031.34810.21992.5344e-041.49820.281513.826780.45170.030316.500091.4954
241251.66264.48970.03030.49901.878815.09380.99994.02670.50000.030316.439491.25290.10193.41580.50000.030316.439491.25291.12920.10282.74370.02560.49881.874015.00431.47560.73507.04590.03030.49891.878815.09431.00001.00004.02670.50000.030316.439491.25290.62300.05120.12593.73661.921021.45111.35950.12361.5615e-041.68980.01781.904717.00560.030316.500091.4954
251260.32723.87010.3935-0.319311.335268.35711.00004.07690.5000-0.250012.000071.97890.09902.87930.4680-0.267312.060172.29971.11860.87484.11860.5000-0.250012.000071.97890.86901.51857.88740.3892-0.328711.330168.32471.00001.00004.07690.5000-0.250012.000071.97891.35111.7080e-061.4988e-050.006815.990988.68650.49291.20890.85433.98810.500016.000088.7228-0.25001688.7228
261271.31996.59760.3788-0.100015.151585.95971.00004.11080.5000-0.075816.121289.97442.1886e-043.50740.4997-0.075816.121289.97441.60110.09227.59400.3788-0.100015.151585.95971.61270.43027.27170.3788-0.100015.151585.95971.00001.00004.11090.5000-0.075816.121289.97441.34104.7227e-074.3497e-060.019916.420691.18070.79171.05371.00634.00900.500016.500091.4954-0.075816.500091.4954
271281.11055.47540.0455-0.49922.727320.11090.99994.02590.5000-0.045516.363690.94920.00593.26320.4997-0.045516.363890.95001.53740.14346.63910.0455-0.49932.727320.11021.37310.53265.06210.0455-0.49932.727320.11081.00001.00004.02590.5000-0.045516.363690.94921.60760.07100.54594.91573.0000138.15631.68390.07640.50027.88190.04552.863624.4078-0.045516.500091.4954
281291.00154.08340.5000-0.053816.061589.35381.27754.13030.3546-0.072814.688483.71480.94744.00540.3480-0.074214.688483.71481.84212.000010.00000.3068-0.228214.118980.85100.93051.03494.00880.5000-0.053816.061589.35381.00001.00004.03830.3546-0.072814.688483.71480.12140.37341.16350.177715.065485.35510.44760.21302.00009.99980.316514.268782.0730-0.053816.250090.1091
291301.36854.17800.0157-0.49780.96888.90460.99994.01630.5000-0.015615.984488.66030.04693.33880.5000-0.015615.984688.66111.48060.13986.96810.0157-0.49810.96888.90411.61250.67014.20980.0157-0.49790.96888.90481.00001.00004.01630.5000-0.015615.984488.66031.57790.10420.64686.88971.000046.05241.66360.07720.52477.66330.01560.984410.3021-0.01561688.7228
301311.76534.67470.0909-0.16675.393939.46570.99994.02760.5000-0.030316.439491.25290.01682.80240.4973-0.030716.442891.26641.59880.09427.51680.0909-0.16675.393939.46571.52930.65937.41290.0909-0.16675.393939.46571.00001.00004.02760.5000-0.030316.439491.25291.60060.08630.66605.94016.0000285.52141.66300.07420.48178.04970.09095.454540.2120-0.030316.500091.4954
311320.99654.08070.5000-0.090915.954589.30140.34693.93800.1201-0.37586.482243.06531.16495.51480.1200-0.37886.484843.06810.48031.46816.73190.1212-0.37506.484843.06810.22111.99949.99700.1061-0.50006.443841.34171.00001.00004.09950.1212-0.37506.484843.06812.5178e-081.1479e-070.43314.57427.030348.75200.57911.49150.98045.35040.12127.030348.7520-0.090916.500091.4954
321330.99724.01120.5000-0.136415.272786.52360.04762.46690.1340-0.500013.907680.52460.89993.98210.3719-0.183414.303082.27200.48540.82460.33970.2110-0.393713.822580.54230.01581.74458.47150.3788-0.180014.303082.27210.08673.4871e-081.35380.1340-0.500013.907680.52467.4038e-071.33866.4444e-060.063415.098585.76324.6123e-061.33424.0143e-050.06378.3208e-0515.098585.7632-0.136416.500091.4954
331340.27843.78970.3565-0.134614.620183.33391.00003.97570.5000-0.084615.784688.23860.00133.48110.4952-0.085515.784888.23940.30470.02154.16170.3549-0.137714.617783.32000.86931.67318.82580.3344-0.179614.539082.93741.00001.00003.97570.5000-0.084615.784688.23861.35594.9393e-074.6904e-060.034316.023289.22670.73501.15870.91224.03840.500016.250090.1091-0.084616.250090.1091
341351.00194.08120.5000-0.069215.938588.85901.27894.11990.3835-0.092615.056885.22410.96844.03890.3788-0.093815.056885.22411.50171.99609.98670.3549-0.149914.762783.81640.86941.06633.99690.5000-0.069215.938588.85901.10461.03524.09000.3835-0.092615.056885.22412.0210e-073.5272e-071.28420.366015.384686.61620.33100.10871.99989.99900.362915.221885.9427-0.069216.250090.1091
351361.30066.65210.1061-0.35715.878840.25561.00004.03770.5000-0.075816.121289.97440.19013.48700.5000-0.075816.121289.97441.58230.09757.43610.1061-0.35715.878840.25561.52400.31116.03260.1061-0.35715.878840.25561.00001.00004.03770.5000-0.075816.121289.97441.53070.08040.39595.19667.0000326.96821.73300.05770.66067.61530.10616.257644.6423-0.075816.500091.4954
361380.99884.01060.5000-0.137913.396675.93351.32804.10720.3571-0.200012.214370.68530.92783.98780.3499-0.204112.214370.68530.33351.58394.51840.3571-0.200012.214370.68530.90861.04514.00160.5000-0.137913.396675.93351.00001.00004.02130.3571-0.200012.214370.68532.6759e-094.9825e-091.34860.420013.396675.93350.49391.55461.10104.06850.362113.396675.9335-0.137914.500080.4051
371400.99854.02330.50000.038516.153889.72411.37934.15350.40060.046115.511487.12190.88533.96920.39450.046815.511487.12190.80871.31106.22110.39520.040015.491287.04080.65201.17023.97870.50000.038516.153889.72411.25261.05954.14280.40060.046115.511487.12190.06773.0698e-071.34220.398315.587187.43911.50331.54521.99979.99870.369215.138885.61210.038516.250090.1091
381411.30416.51540.4768-0.041215.616986.80301.00004.01920.5000-0.039715.650886.93930.01243.59950.4888-0.040615.651586.94221.84461.99989.99910.4265-0.086215.382885.84581.47200.59496.95540.4768-0.041215.616986.80301.00001.00004.01920.5000-0.039715.650886.93930.81820.24850.89585.0677e-0715.750087.33661.68310.07290.51117.45090.476215.714387.1936-0.039715.750087.3365
391421.30286.51490.4242-0.071415.878888.97921.00004.05140.5000-0.060616.257690.52330.04481.74380.4684-0.064816.270090.57391.57430.09007.24950.4242-0.071415.878888.97921.40400.58117.42040.4242-0.071415.878888.97921.000014.05140.5000-0.060616.257690.52331.33051.0211e-061.1015e-050.019916.421191.18060.69111.07011.06754.00150.500016.500091.4954-0.060616.500091.4954
401431.30486.55310.2633-0.212211.742470.41171.00004.08100.5000-0.115415.384686.61625.3342e-081.99970.2549-0.226315.384686.61621.59820.09407.67070.2633-0.212211.742470.41171.89680.52884.89410.2633-0.212211.742470.41171.00001.00004.08100.5000-0.115415.384686.61620.04506.3723e-094.8005e-083.387714.229882.13170.36621.53651.14054.00270.500016.250090.1092-0.115416.250090.1091
411441.36534.16870.0162-0.49790.96778.84010.99994.02700.5000-0.023815.714387.19360.01592.74260.4974-0.024715.717887.20761.49620.13526.63200.0162-0.49810.96778.83951.60520.67894.20840.0162-0.49800.96778.84031.00001.00004.02700.5000-0.023815.714387.19361.59730.08040.58984.98271.000046.05371.66490.08060.51587.72450.01590.984110.2703-0.023815.750087.3365
421450.15553.78590.4753-0.139414.442981.66571.00004.03030.5000-0.129014.467781.77410.79243.72150.4981-0.129314.471381.78941.84411.78139.53240.4296-0.192614.248280.79150.69631.99969.99820.4449-0.178914.336581.19591.00001.00004.03030.5000-0.129014.467781.77410.93830.02350.17260.003815.517386.01940.79101.26911.04304.29080.500015.500085.9503-0.129015.500085.9503
431461.26396.27200.4697-0.145215.212186.26151.00004.01580.5000-0.136415.272786.52360.00623.54850.4965-0.137315.272986.52431.62591.66098.96480.4761-0.154715.244986.40501.42800.65116.91700.4697-0.145215.212186.26151.00001.00004.01580.5000-0.136415.272786.52360.76750.60070.96047.2121e-0716.500091.49551.65500.08150.51417.80160.469716.439491.2529-0.136416.500091.4954
441470.98624.01130.5000-0.007716.246290.09381.01334.99608.5216e-059.3067e-044.7218e-070.01111.52657.63464.5476e-05-0.00271.9064e-070.00700.56351.33525.83444.3832e-05-0.00211.3269e-070.00580.05211.69388.22763.9041e-050.03231.6941e-070.00651.00021.00014.99688.5243e-050.00144.7237e-070.01110.64021.28840.71744.85621.8534e-087.2232e-040.59311.26911.06704.34084.3729e-051.3274e-070.0059-0.007716.250090.1091
451481.34554.20340.0909-0.49964.909131.29801.00004.08180.5000-0.090915.954589.30145.8531e-043.07480.4961-0.091615.954789.30221.50160.13726.68830.0909-0.49974.909131.29741.29430.66526.04900.0909-0.49974.909131.29741.00001.00004.08190.5000-0.090915.954589.30141.62050.08680.52705.22786.0000276.31131.69540.07260.49207.90970.09095.454540.2120-0.090916.500091.4954
461490.51592.75403.2517e-06-0.13370.94477.85350.99994.01630.50000.015216.484991.43480.02331.48390.49740.015216.487791.44641.99970.36359.99862.8582e-05-0.31410.50132.88250.87761.05864.06825.0613e-050.05210.94447.73051.00001.00004.01630.50000.015216.484991.43481.99970.36355.5497e-049.99860.50132.87881.99990.36351.1126e-049.99972.5650e-060.50132.87900.015216.500091.4954
471500.99864.01570.5000-0.112914.709782.76151.30924.15620.3260-0.164512.835474.67450.91123.99510.3167-0.169412.835474.67450.50200.85573.93000.3174-0.181012.807174.44960.90561.04664.00110.5000-0.112914.709782.76151.00001.00004.02520.3260-0.164512.835474.67455.1432e-071.34221.4554e-050.087213.243876.59460.58441.29761.11686.10300.322613.548477.9711-0.112915.500085.9503
481511.34196.68100.2214-0.17749.728659.61840.99994.01960.5000-0.069014.224179.29810.20813.52290.5000-0.069014.224179.29811.57870.08517.47170.2214-0.17749.728659.61841.54400.64807.22280.2214-0.17749.728659.61841.00001.00004.01970.5000-0.069014.224179.29810.34071.1267e-073.7153e-070.634011.363967.82361.66710.07240.47118.14090.224110.086261.7231-0.069014.500080.4051
491520.28073.86460.1759-0.098210.378765.07760.99994.01670.5000-0.015216.484991.43480.07533.46030.5000-0.015116.485091.43551.54000.08897.36380.1970-0.038510.424265.39890.90521.57747.94160.1727-0.114010.359664.95541.00001.00004.01670.5000-0.015216.484991.43481.34988.4464e-076.7136e-060.241311.357372.98720.95592.6209e-071.53147.44750.187510.423665.3969-0.015216.500091.4954
501530.99854.02270.50000.045516.363690.94921.03324.98780.40910.055615.818288.73640.93333.97850.40450.056215.818288.73640.46231.43666.39940.40910.055615.818288.73640.80661.10143.99260.50000.045516.363690.94920.99880.99915.00010.40910.055615.818288.73643.7629e-082.3571e-090.10643.455015.954589.30140.52631.50881.13914.35970.409115.954589.30140.045516.500091.4954
511541.32006.60390.0303-2.5112e-071.939417.92480.99993.98480.50001.1207e-0816.500091.49540.04503.48780.4992-2.8185e-0516.503791.51021.58540.09547.40020.03031.1926e-071.939417.92481.52640.68677.30450.0303-2.5745e-061.939417.92481.00001.00003.98240.5000-8.7941e-0916.500091.49541.56920.08820.49105.58342.000096.70941.67800.07730.51247.83430.03031.939417.9248016.500091.4954
521551.29746.48830.06060.25003.697029.06840.99994.02710.50000.030316.439491.25290.01512.58550.49710.030916.442791.26622.00000.362710.00000.04690.17753.347226.66151.53220.69757.34060.06060.25003.697029.06841.00001.00004.02710.50000.030316.439491.25292.00000.36352.7837e-059.99993.5013150.24412.00000.36344.4299e-059.99990.04633.364827.09140.030316.500091.4954
531560.98634.01150.5000-0.015216.484991.43481.33584.18760.0455-0.16672.848524.05211.24064.36730.0414-0.18302.848524.05210.48710.97132.13340.0310-0.38742.766421.66900.01791.77168.66530.0455-0.16672.848524.05211.00001.00004.06680.0455-0.16672.848524.05210.60211.28221.3695e-051.27032.860852.66510.61551.27151.0495e-051.62290.03372.801023.5251-0.015216.500091.4954
541570.15541.48750.1110-0.500013.572778.74011.00004.05180.5000-0.053816.061589.35380.01453.28780.4932-0.054616.065289.36870.94270.26190.43320.1366-0.500013.348677.70720.39870.98651.81860.1660-0.500013.289777.31061.00001.00004.05180.5000-0.053816.061589.35381.33832.4066e-063.4647e-060.104914.404182.64562.00000.36137.6701e-0610.00000.317714.161381.5627-0.053816.250090.1091
551581.00384.05270.5000-0.060616.257690.52340.42013.88530.4848-0.062516.242490.46160.93994.74120.4845-0.062516.242490.46180.52331.46236.30150.4848-0.062516.242490.46181.10200.93884.05160.5000-0.060616.257690.52331.00001.00004.04990.4848-0.062516.242490.46180.77821.00711.20645.2343e-0816.500091.49540.51551.34961.11184.17440.484816.484891.4348-0.060616.500091.4954
561600.98904.01100.50000.023116.215489.97061.33484.12840.15340.09268.409155.40651.07744.05970.14460.09838.409155.40650.45241.46046.41630.15340.09268.409155.40650.21921.99999.99940.14130.05768.384355.24181.00001.00004.03580.15340.09268.409155.40650.00993.3245e-080.38535.06748.450555.74430.50881.55330.99814.50660.15388.461555.81200.023116.250090.1091
571610.99334.07410.5000-0.161514.553883.20131.20484.10660.4129-0.197214.060680.97670.92373.99030.4084-0.199414.060680.97670.34901.58334.74390.4129-0.197214.060680.97670.01971.80938.95450.4129-0.197214.060680.97671.13810.98754.06850.4129-0.197214.060680.97676.9204e-081.6940e-071.34330.254415.784688.23860.47251.42551.07894.45430.415415.784688.2386-0.161516.250090.1091

Let's glance at the trial level data for these alternative models.

altTrialData
altTrialData = 3704×17 table 
 SubjectIDa0b0a1b1a2b2Chose1alphaOnly_Prob1deltaOnly_Prob1rhoOnly_Prob1ad_Prob1ar_Prob1dr_Prob1noEpsilon_Prob1noGamma_Prob1gammaOnly_Prob1
110116514781310.45450.51520.51520.51520.45450.51520.56060.56060.5000
210116151215200.45450.39390.39390.39390.45450.39390.43940.43940.5000
310116315451400.45450.51520.51520.51520.45450.51520.56060.56060.5000
4101516111071400.45450.39390.39390.39390.45450.39390.43940.43940.5000
51011528913400.45450.39390.39390.39390.45450.39390.43940.43940.5000
610151671412910.45450.51520.51520.51520.45450.51520.56060.56060.5000
710111512421400.45450.39390.39390.39390.45450.39390.43940.43940.5000
810121510741300.45450.39390.39390.39390.45450.39390.43940.43940.5000
910115381014400.45450.39390.39390.39390.45450.39390.43940.43940.5000
1010131510851300.45450.39390.39390.39390.45450.39390.43940.43940.5000
1110115161014200.45450.39390.39390.39390.45450.39390.43940.43940.5000
1210131511741410.45450.39390.39390.39390.45450.39390.43940.43940.5000
1310115311781000.45450.51520.51520.51520.45450.51520.56060.56060.5000
1410114213341200.45450.51520.51520.51520.45450.51520.56060.56060.5000
1510115271014300.45450.39390.39390.39390.45450.39390.43940.43940.5000
1610131612741500.45450.39390.39390.39390.45450.39390.43940.43940.5000
1710115211671000.45450.51520.51520.51520.45450.51520.56060.56060.5000
1810121651311700.45450.51520.51520.51520.45450.51520.56060.56060.5000
1910121641412610.45450.51520.51520.51520.45450.51520.56060.56060.5000
201011166119810.45450.51520.51520.51520.45450.51520.56060.56060.5000
2110131651412710.45450.51520.51520.51520.45450.51520.56060.56060.5000
2210116213561200.45450.51520.51520.51520.45450.51520.56060.56060.5000
2310115314451310.45450.51520.51520.51520.45450.51520.56060.56060.5000
2410131611851400.45450.39390.39390.39390.45450.39390.43940.43940.5000
2510131541412610.45450.51520.51520.51520.45450.51520.56060.56060.5000
2610131661311810.45450.51520.51520.51520.45450.51520.56060.56060.5000
2710116391014500.45450.39390.39390.39390.45450.39390.43940.43940.5000
2810115212561110.45450.51520.51520.51520.45450.51520.56060.56060.5000
291011159741200.45450.39390.39390.39390.45450.39390.43940.43940.5000
301011517913310.45450.39390.39390.39390.45450.39390.43940.43940.5000
3110121613531510.45450.39390.39390.39390.45450.39390.43940.43940.5000
3210121531412500.45450.51520.51520.51520.45450.51520.56060.56060.5000
3310116515671400.45450.51520.51520.51520.45450.51520.56060.56060.5000
3410115312671100.45450.51520.51520.51520.45450.51520.56060.56060.5000
3510116271115310.45450.39390.39390.39390.45450.39390.43940.43940.5000
3610151661513800.45450.51520.51520.51520.45450.51520.56060.56060.5000
3710115114231300.45450.51520.51520.51520.45450.51520.56060.56060.5000
381011327812300.45450.39390.39390.39390.45450.39390.43940.43940.5000
3910141671311910.45450.51520.51520.51520.45450.51520.56060.56060.5000
4010121611741400.45450.39390.39390.39390.45450.39390.43940.43940.5000
4110116211781010.45450.51520.51520.51520.45450.51520.56060.56060.5000
4210111511521400.45450.39390.39390.39390.45450.39390.43940.43940.5000
4310116381115410.45450.39390.39390.39390.45450.39390.43940.43940.5000
4410131610961300.45450.39390.39390.39390.45450.39390.43940.43940.5000
451012149741200.45450.39390.39390.39390.45450.39390.43940.43940.5000
4610121410631310.45450.39390.39390.39390.45450.39390.43940.43940.5000
4710116215341410.45450.51520.51520.51520.45450.51520.56060.56060.5000
4810131641513610.45450.51520.51520.51520.45450.51520.56060.56060.5000
4910151612961510.45450.39390.39390.39390.45450.39390.43940.43940.5000
5010131551311700.45450.51520.51520.51520.45450.51520.56060.56060.5000
51101165101115610.45450.39390.39390.39390.45450.39390.43940.43940.5000
5210116171014310.45450.39390.39390.39390.45450.39390.43940.43940.5000
5310121541311600.45450.51520.51520.51520.45450.51520.56060.56060.5000
5410111613421500.45450.39390.39390.39390.45450.39390.43940.43940.5000
5510116281014410.45450.39390.39390.39390.45450.39390.43940.43940.5000
5610121612631510.45450.39390.39390.39390.45450.39390.43940.43940.5000
5710115313561200.45450.51520.51520.51520.45450.51520.56060.56060.5000
5810121661210810.45450.51520.51520.51520.45450.51520.56060.56060.5000
5910116212671110.45450.51520.51520.51520.45450.51520.56060.56060.5000
6010116513891210.45450.51520.51520.51520.45450.51520.56060.56060.5000
6110115214341300.45450.51520.51520.51520.45450.51520.56060.56060.5000
6210116161115210.45450.39390.39390.39390.45450.39390.43940.43940.5000
631011427913300.45450.39390.39390.39390.45450.39390.43940.43940.5000
6410111531311510.45450.51520.51520.51520.45450.51520.56060.56060.5000
6510115213451200.45450.51520.51520.51520.45450.51520.56060.56060.5000
6610121511631410.45450.39390.39390.39390.45450.39390.43940.43940.5000
6710216514781300.06060.42420.42420.05140.06060.42420.03000.13640.5000
6810216151215200.78790.42420.42420.78780.78790.42421.00000.86360.5000
6910216315451410.06060.42420.42420.12580.06060.42420.13060.13640.5000
70102516111071400.78790.42420.42420.78780.78790.42421.00000.86360.5000
711021528913400.78790.42420.42420.78780.78790.42421.00000.86360.5000
7210251671412900.06060.42420.42420.03910.06060.42425.9968e-050.13640.5000
7310211512421410.78790.42420.42420.78770.78790.42420.99960.86360.5000
7410221510741300.78790.42420.42420.78780.78790.42421.00000.86360.5000
7510215381014410.78790.42420.42420.78780.78790.42421.00000.86360.5000
7610231510851310.78790.42420.42420.78780.78790.42421.00000.86360.5000
7710215161014210.78790.42420.42420.78780.78790.42421.00000.86360.5000
7810231511741410.78790.42420.42420.78780.78790.42421.00000.86360.5000
7910215311781000.06060.42420.42420.04170.06060.42420.00940.13640.5000
8010214213341200.06060.42420.42420.09340.06060.42420.09190.13640.5000
8110215271014310.78790.42420.42420.78780.78790.42421.00000.86360.5000
8210231612741510.78790.42420.42420.78780.78790.42421.00000.86360.5000
8310215211671000.06060.42420.42420.04350.06060.42420.01380.13640.5000
8410221651311700.06060.42420.42420.03910.06060.42428.9042e-050.13640.5000
8510221641412600.06060.42420.42420.03910.06060.42421.9630e-040.13640.5000
861021166119800.06060.42420.42420.03910.06060.42422.7199e-050.13640.5000
8710231651412710.06060.42420.42420.03910.06060.42421.3221e-040.13640.5000
8810216213561200.06060.42420.42420.05940.06060.42420.04390.13640.5000
8910215314451300.06060.42420.42420.09340.06060.42420.09190.13640.5000
9010231611851410.78790.42420.42420.78780.78790.42421.00000.86360.5000
9110231541412600.06060.42420.42420.03910.06060.42421.9630e-040.13640.5000
9210231661311800.06060.42420.42420.03910.06060.42425.9968e-050.13640.5000
9310216391014510.78790.42420.42420.78780.78790.42421.00000.86360.5000
9410215212561100.06060.42420.42420.05140.06060.42420.03000.13640.5000
951021159741210.78790.42420.42420.78780.78790.42421.00000.86360.5000
961021517913310.78790.42420.42420.78780.78790.42421.00000.86360.5000
9710221613531510.78790.42420.42420.78770.78790.42420.99960.86360.5000
9810221531412500.06060.42420.42420.03910.06060.42422.9146e-040.13640.5000
9910216515671400.06060.42420.42420.07250.06060.42420.06380.13640.5000
10010215312671100.06060.42420.42420.04650.06060.42420.02040.13640.5000
⋮

	SubjectID	a0	b0	a1	b1	a2	b2	Chose1	alphaOnly_Prob1	deltaOnly_Prob1	rhoOnly_Prob1	ad_Prob1	ar_Prob1	dr_Prob1	noEpsilon_Prob1	noGamma_Prob1	gammaOnly_Prob1
1	101	16	5	14	7	8	13	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
2	101	16	1	5	12	15	2	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
3	101	16	3	15	4	5	14	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
4	101	5	16	11	10	7	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
5	101	15	2	8	9	13	4	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
6	101	5	16	7	14	12	9	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
7	101	1	15	12	4	2	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
8	101	2	15	10	7	4	13	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
9	101	15	3	8	10	14	4	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
10	101	3	15	10	8	5	13	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
11	101	15	1	6	10	14	2	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
12	101	3	15	11	7	4	14	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
13	101	15	3	11	7	8	10	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
14	101	14	2	13	3	4	12	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
15	101	15	2	7	10	14	3	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
16	101	3	16	12	7	4	15	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
17	101	15	2	11	6	7	10	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
18	101	2	16	5	13	11	7	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
19	101	2	16	4	14	12	6	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
20	101	1	16	6	11	9	8	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
21	101	3	16	5	14	12	7	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
22	101	16	2	13	5	6	12	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
23	101	15	3	14	4	5	13	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
24	101	3	16	11	8	5	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
25	101	3	15	4	14	12	6	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
26	101	3	16	6	13	11	8	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
27	101	16	3	9	10	14	5	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
28	101	15	2	12	5	6	11	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
29	101	1	15	9	7	4	12	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
30	101	15	1	7	9	13	3	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
31	101	2	16	13	5	3	15	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
32	101	2	15	3	14	12	5	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
33	101	16	5	15	6	7	14	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
34	101	15	3	12	6	7	11	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
35	101	16	2	7	11	15	3	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
36	101	5	16	6	15	13	8	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
37	101	15	1	14	2	3	13	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
38	101	13	2	7	8	12	3	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
39	101	4	16	7	13	11	9	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
40	101	2	16	11	7	4	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
41	101	16	2	11	7	8	10	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
42	101	1	15	11	5	2	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
43	101	16	3	8	11	15	4	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
44	101	3	16	10	9	6	13	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
45	101	2	14	9	7	4	12	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
46	101	2	14	10	6	3	13	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
47	101	16	2	15	3	4	14	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
48	101	3	16	4	15	13	6	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
49	101	5	16	12	9	6	15	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
50	101	3	15	5	13	11	7	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
51	101	16	5	10	11	15	6	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
52	101	16	1	7	10	14	3	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
53	101	2	15	4	13	11	6	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
54	101	1	16	13	4	2	15	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
55	101	16	2	8	10	14	4	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
56	101	2	16	12	6	3	15	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
57	101	15	3	13	5	6	12	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
58	101	2	16	6	12	10	8	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
59	101	16	2	12	6	7	11	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
60	101	16	5	13	8	9	12	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
61	101	15	2	14	3	4	13	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
62	101	16	1	6	11	15	2	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
63	101	14	2	7	9	13	3	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
64	101	1	15	3	13	11	5	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
65	101	15	2	13	4	5	12	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000
66	101	2	15	11	6	3	14	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000
67	102	16	5	14	7	8	13	0	0.0606	0.4242	0.4242	0.0514	0.0606	0.4242	0.0300	0.1364	0.5000
68	102	16	1	5	12	15	2	0	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
69	102	16	3	15	4	5	14	1	0.0606	0.4242	0.4242	0.1258	0.0606	0.4242	0.1306	0.1364	0.5000
70	102	5	16	11	10	7	14	0	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
71	102	15	2	8	9	13	4	0	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
72	102	5	16	7	14	12	9	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	5.9968e-05	0.1364	0.5000
73	102	1	15	12	4	2	14	1	0.7879	0.4242	0.4242	0.7877	0.7879	0.4242	0.9996	0.8636	0.5000
74	102	2	15	10	7	4	13	0	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
75	102	15	3	8	10	14	4	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
76	102	3	15	10	8	5	13	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
77	102	15	1	6	10	14	2	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
78	102	3	15	11	7	4	14	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
79	102	15	3	11	7	8	10	0	0.0606	0.4242	0.4242	0.0417	0.0606	0.4242	0.0094	0.1364	0.5000
80	102	14	2	13	3	4	12	0	0.0606	0.4242	0.4242	0.0934	0.0606	0.4242	0.0919	0.1364	0.5000
81	102	15	2	7	10	14	3	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
82	102	3	16	12	7	4	15	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
83	102	15	2	11	6	7	10	0	0.0606	0.4242	0.4242	0.0435	0.0606	0.4242	0.0138	0.1364	0.5000
84	102	2	16	5	13	11	7	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	8.9042e-05	0.1364	0.5000
85	102	2	16	4	14	12	6	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	1.9630e-04	0.1364	0.5000
86	102	1	16	6	11	9	8	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	2.7199e-05	0.1364	0.5000
87	102	3	16	5	14	12	7	1	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	1.3221e-04	0.1364	0.5000
88	102	16	2	13	5	6	12	0	0.0606	0.4242	0.4242	0.0594	0.0606	0.4242	0.0439	0.1364	0.5000
89	102	15	3	14	4	5	13	0	0.0606	0.4242	0.4242	0.0934	0.0606	0.4242	0.0919	0.1364	0.5000
90	102	3	16	11	8	5	14	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
91	102	3	15	4	14	12	6	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	1.9630e-04	0.1364	0.5000
92	102	3	16	6	13	11	8	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	5.9968e-05	0.1364	0.5000
93	102	16	3	9	10	14	5	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
94	102	15	2	12	5	6	11	0	0.0606	0.4242	0.4242	0.0514	0.0606	0.4242	0.0300	0.1364	0.5000
95	102	1	15	9	7	4	12	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
96	102	15	1	7	9	13	3	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000
97	102	2	16	13	5	3	15	1	0.7879	0.4242	0.4242	0.7877	0.7879	0.4242	0.9996	0.8636	0.5000
98	102	2	15	3	14	12	5	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	2.9146e-04	0.1364	0.5000
99	102	16	5	15	6	7	14	0	0.0606	0.4242	0.4242	0.0725	0.0606	0.4242	0.0638	0.1364	0.5000
100	102	15	3	12	6	7	11	0	0.0606	0.4242	0.4242	0.0465	0.0606	0.4242	0.0204	0.1364	0.5000
⋮

Now we can compute BIC for these models

n = 65;
altSubjectData.BIC_M1 = altSubjectData.Deviance_M1 + log(n) * 4;
altSubjectData.BIC_M2 = altSubjectData.Deviance_M2 + log(n) * 4;
altSubjectData.BIC_M3 = altSubjectData.Deviance_M3 + log(n) * 4;
altSubjectData.BIC_M4 = altSubjectData.Deviance_M4 + log(n) * 5;
altSubjectData.BIC_M5 = altSubjectData.Deviance_M5 + log(n) * 5;
altSubjectData.BIC_M6 = altSubjectData.Deviance_M6 + log(n) * 5;
altSubjectData.BIC_M7 = altSubjectData.Deviance_M7 + log(n) * 4;
altSubjectData.BIC_M8 = altSubjectData.Deviance_M8 + log(n) * 5;
altSubjectData.BIC_M9 = altSubjectData.Deviance_M9 + log(n) * 1;

And now we can compare the BIC of all models

modelBIC = [sum(subjectData.BIC), sum(altSubjectData.BIC_M1), sum(altSubjectData.BIC_M2), sum(altSubjectData.BIC_M3), sum(altSubjectData.BIC_M4), ...
            sum(altSubjectData.BIC_M5), sum(altSubjectData.BIC_M6), sum(altSubjectData.BIC_M7), sum(altSubjectData.BIC_M8), sum(altSubjectData.BIC_M9)];
 
[minBIC, idx] = min(modelBIC);
disp(idx);
     5

So model 4, which has everything except Rho

2.4 Validate the Best Model

First, let’s assess model performance at a basic level: we can look at prediction accuracy to begin

sum(altTrialData.Chose1 == round(altTrialData.ad_Prob1))/height(altTrialData)
ans = 0.7916

Next, we can proceed to look at the distribution of model accuracy across participants:

[f, xi] = ksdensity(altSubjectData.BIC_M4);

plot(xi, f);

title('Density of BIC M4');

xlabel('BIC M4');

ylabel('Density');

Let’s look at the bottom 25% of model performance:

Q3 = prctile(altSubjectData.BIC_M4, 75);

worstExplained = find(altSubjectData.BIC_M4 > Q3);

filteredData = altTrialData(ismember(altTrialData.SubjectID, altSubjectData.SubjectID(worstExplained)), :);

x_diff = filteredData.a0 - filteredData.a1;

y_diff = filteredData.Chose1 - filteredData.ad_Prob1;

figure;

hold on;

uniqueSubjects = unique(filteredData.SubjectID);

colors = lines(length(uniqueSubjects));

for i = 1:length(uniqueSubjects)

idx = filteredData.SubjectID == uniqueSubjects(i);

scatter(x_diff(idx), y_diff(idx), [], 'Color', colors(i));

lsline;

end

hold off;

xlabel('a0 - a1');

ylabel('Chose1 - ad\_Prob1');

Nothing systematic here

x_diff = filteredData.a0 - filteredData.a2;

figure;

hold on;

uniqueSubjects = unique(filteredData.SubjectID);

colors = lines(length(uniqueSubjects));

for i = 1:length(uniqueSubjects)

idx = filteredData.SubjectID == uniqueSubjects(i);

scatter(x_diff(idx), y_diff(idx), [], 'Color', colors(i));

lsline;

end

hold off;

xlabel('a0 - a2');

ylabel('Chose1 - ad\_Prob1');

Systematically overpredicting choosing 1 when player A has more and undepredicted when player A has less. This suggests a bias that only seems to apply when looking at choice 2 because we don't see this on choice 1.

x_diff = filteredData.b0 - filteredData.b1;

figure;

hold on;

uniqueSubjects = unique(filteredData.SubjectID);

colors = lines(length(uniqueSubjects));

for i = 1:length(uniqueSubjects)

idx = filteredData.SubjectID == uniqueSubjects(i);

scatter(x_diff(idx), y_diff(idx), [], 'Color', colors(i));

lsline;

end

hold off;

xlabel('b0 - b1');

ylabel('Chose1 - ad\_Prob1');

This is fine

x_diff = filteredData.b0 - filteredData.b2;

figure;

hold on;

uniqueSubjects = unique(filteredData.SubjectID);

colors = lines(length(uniqueSubjects));

for i = 1:length(uniqueSubjects)

idx = filteredData.SubjectID == uniqueSubjects(i);

scatter(x_diff(idx), y_diff(idx), [], 'Color', colors(i));

lsline;

end

hold off;

xlabel('b0 - b2');

ylabel('Chose1 - ad\_Prob1');

Same issue as before - so it seems that these people have a preference/bias for player A when player B has more, but this only applies to Choice 1. Not very useful information, but we're dealing with a trialset that is probably too small for the number of participants we have. Anyways, we can't do anything with this information, we can just bear it in mind as we proceed to check assumptions: first linearity (we’ll do this across both choices)

figure;

hold on;

groups = altTrialData.a0 < altTrialData.b0;

uniqueGroups = unique(groups);

titles = ["a0 < b0", "a0 > b0"];

colors = ["r*", "b*"];

for i = 1:length(uniqueGroups)

idx = groups == uniqueGroups(i);

plot(altTrialData.ad_Prob1(idx), altTrialData.Chose1(idx), colors(i));

end

lsline;

xlabel('ad\_Prob1');

ylabel('Chose1');

legend(titles)

hold off

This isn't great, but it's fine. Second, normality of error:

residuals = altTrialData.ad_Prob1 - altTrialData.Chose1;

residual_std = std(residuals);

normvals = normrnd(0, residual_std, 1000, 1);

[f, xi] = ksdensity(residuals, 'Bandwidth', residual_std);

[f_norm, xi_norm] = ksdensity(normvals, 'Bandwidth', residual_std);

figure;

hold on;

plot(xi, f, 'LineWidth', 2, 'Color', 'b');

plot(xi_norm, f_norm, 'LineWidth', 2, 'Color', 'r');

xlabel('Residuals');

ylabel('Density');

legend({'Actual', 'Predicted'});

hold off;

Looks very good. Third we can examine homoscedasticity:

x_vals = altTrialData.a1 - altTrialData.a2;

y_vals = altTrialData.ad_Prob1 - altTrialData.Chose1;

scatter(x_vals, y_vals);

hold on;

lsline;

mdl = fitlm(x_vals, y_vals);

[x_sorted, sortIdx] = sort(x_vals);

[y_pred, y_ci] = predict(mdl, x_sorted);

fill([x_sorted; flipud(x_sorted)], [y_ci(:,1); flipud(y_ci(:,2))], 'k', ...

'FaceAlpha', 0.2, 'EdgeColor', 'none');

xlabel('a1 - a2');

ylabel('ad\_Prob1 - Chose1');

hold off;

Nice constant variance cloud across all X values. And finally independence of error:

figure;

hold on;

groups = altTrialData.a0 < altTrialData.b0;

uniqueGroups = unique(groups);

titles = ["a0 < b0", "a0 > b0"];

colors = ["r*", "b*"];

for i = 1:length(uniqueGroups)

idx = groups == uniqueGroups(i);

x = altTrialData.a1(idx) - altTrialData.a2(idx);

diff = altTrialData.Chose1(idx) - altTrialData.ad_Prob1(idx);

plot(x, diff, colors(i));

end

lsline;

xlabel('a1 - a2');

ylabel('Chose1 - ad\_Prob1');

legend(titles)

hold off

The model seems to have a slight bias, under-predicting the likelihood of choosing option 1 when Player A has less than Player B and Choice 2 is better for player A. But this seems to not be such a big issue.

Let’s assess the independence: i.e. the extent to which our model captures all differences in choice behavior between different people:

altTrialData.a0greaterThanb0 = altTrialData.a0 > altTrialData.b0
altTrialData = 3704×18 table 
 SubjectIDa0b0a1b1a2b2Chose1alphaOnly_Prob1deltaOnly_Prob1rhoOnly_Prob1ad_Prob1ar_Prob1dr_Prob1noEpsilon_Prob1noGamma_Prob1gammaOnly_Prob1a0greaterThanb0
110116514781310.45450.51520.51520.51520.45450.51520.56060.56060.50001
210116151215200.45450.39390.39390.39390.45450.39390.43940.43940.50001
310116315451400.45450.51520.51520.51520.45450.51520.56060.56060.50001
4101516111071400.45450.39390.39390.39390.45450.39390.43940.43940.50000
51011528913400.45450.39390.39390.39390.45450.39390.43940.43940.50001
610151671412910.45450.51520.51520.51520.45450.51520.56060.56060.50000
710111512421400.45450.39390.39390.39390.45450.39390.43940.43940.50000
810121510741300.45450.39390.39390.39390.45450.39390.43940.43940.50000
910115381014400.45450.39390.39390.39390.45450.39390.43940.43940.50001
1010131510851300.45450.39390.39390.39390.45450.39390.43940.43940.50000
1110115161014200.45450.39390.39390.39390.45450.39390.43940.43940.50001
1210131511741410.45450.39390.39390.39390.45450.39390.43940.43940.50000
1310115311781000.45450.51520.51520.51520.45450.51520.56060.56060.50001
1410114213341200.45450.51520.51520.51520.45450.51520.56060.56060.50001
1510115271014300.45450.39390.39390.39390.45450.39390.43940.43940.50001
1610131612741500.45450.39390.39390.39390.45450.39390.43940.43940.50000
1710115211671000.45450.51520.51520.51520.45450.51520.56060.56060.50001
1810121651311700.45450.51520.51520.51520.45450.51520.56060.56060.50000
1910121641412610.45450.51520.51520.51520.45450.51520.56060.56060.50000
201011166119810.45450.51520.51520.51520.45450.51520.56060.56060.50000
2110131651412710.45450.51520.51520.51520.45450.51520.56060.56060.50000
2210116213561200.45450.51520.51520.51520.45450.51520.56060.56060.50001
2310115314451310.45450.51520.51520.51520.45450.51520.56060.56060.50001
2410131611851400.45450.39390.39390.39390.45450.39390.43940.43940.50000
2510131541412610.45450.51520.51520.51520.45450.51520.56060.56060.50000
2610131661311810.45450.51520.51520.51520.45450.51520.56060.56060.50000
2710116391014500.45450.39390.39390.39390.45450.39390.43940.43940.50001
2810115212561110.45450.51520.51520.51520.45450.51520.56060.56060.50001
291011159741200.45450.39390.39390.39390.45450.39390.43940.43940.50000
301011517913310.45450.39390.39390.39390.45450.39390.43940.43940.50001
3110121613531510.45450.39390.39390.39390.45450.39390.43940.43940.50000
3210121531412500.45450.51520.51520.51520.45450.51520.56060.56060.50000
3310116515671400.45450.51520.51520.51520.45450.51520.56060.56060.50001
3410115312671100.45450.51520.51520.51520.45450.51520.56060.56060.50001
3510116271115310.45450.39390.39390.39390.45450.39390.43940.43940.50001
3610151661513800.45450.51520.51520.51520.45450.51520.56060.56060.50000
3710115114231300.45450.51520.51520.51520.45450.51520.56060.56060.50001
381011327812300.45450.39390.39390.39390.45450.39390.43940.43940.50001
3910141671311910.45450.51520.51520.51520.45450.51520.56060.56060.50000
4010121611741400.45450.39390.39390.39390.45450.39390.43940.43940.50000
4110116211781010.45450.51520.51520.51520.45450.51520.56060.56060.50001
4210111511521400.45450.39390.39390.39390.45450.39390.43940.43940.50000
4310116381115410.45450.39390.39390.39390.45450.39390.43940.43940.50001
4410131610961300.45450.39390.39390.39390.45450.39390.43940.43940.50000
451012149741200.45450.39390.39390.39390.45450.39390.43940.43940.50000
4610121410631310.45450.39390.39390.39390.45450.39390.43940.43940.50000
4710116215341410.45450.51520.51520.51520.45450.51520.56060.56060.50001
4810131641513610.45450.51520.51520.51520.45450.51520.56060.56060.50000
4910151612961510.45450.39390.39390.39390.45450.39390.43940.43940.50000
5010131551311700.45450.51520.51520.51520.45450.51520.56060.56060.50000
51101165101115610.45450.39390.39390.39390.45450.39390.43940.43940.50001
5210116171014310.45450.39390.39390.39390.45450.39390.43940.43940.50001
5310121541311600.45450.51520.51520.51520.45450.51520.56060.56060.50000
5410111613421500.45450.39390.39390.39390.45450.39390.43940.43940.50000
5510116281014410.45450.39390.39390.39390.45450.39390.43940.43940.50001
5610121612631510.45450.39390.39390.39390.45450.39390.43940.43940.50000
5710115313561200.45450.51520.51520.51520.45450.51520.56060.56060.50001
5810121661210810.45450.51520.51520.51520.45450.51520.56060.56060.50000
5910116212671110.45450.51520.51520.51520.45450.51520.56060.56060.50001
6010116513891210.45450.51520.51520.51520.45450.51520.56060.56060.50001
6110115214341300.45450.51520.51520.51520.45450.51520.56060.56060.50001
6210116161115210.45450.39390.39390.39390.45450.39390.43940.43940.50001
631011427913300.45450.39390.39390.39390.45450.39390.43940.43940.50001
6410111531311510.45450.51520.51520.51520.45450.51520.56060.56060.50000
6510115213451200.45450.51520.51520.51520.45450.51520.56060.56060.50001
6610121511631410.45450.39390.39390.39390.45450.39390.43940.43940.50000
6710216514781300.06060.42420.42420.05140.06060.42420.03000.13640.50001
6810216151215200.78790.42420.42420.78780.78790.42421.00000.86360.50001
6910216315451410.06060.42420.42420.12580.06060.42420.13060.13640.50001
70102516111071400.78790.42420.42420.78780.78790.42421.00000.86360.50000
711021528913400.78790.42420.42420.78780.78790.42421.00000.86360.50001
7210251671412900.06060.42420.42420.03910.06060.42425.9968e-050.13640.50000
7310211512421410.78790.42420.42420.78770.78790.42420.99960.86360.50000
7410221510741300.78790.42420.42420.78780.78790.42421.00000.86360.50000
7510215381014410.78790.42420.42420.78780.78790.42421.00000.86360.50001
7610231510851310.78790.42420.42420.78780.78790.42421.00000.86360.50000
7710215161014210.78790.42420.42420.78780.78790.42421.00000.86360.50001
7810231511741410.78790.42420.42420.78780.78790.42421.00000.86360.50000
7910215311781000.06060.42420.42420.04170.06060.42420.00940.13640.50001
8010214213341200.06060.42420.42420.09340.06060.42420.09190.13640.50001
8110215271014310.78790.42420.42420.78780.78790.42421.00000.86360.50001
8210231612741510.78790.42420.42420.78780.78790.42421.00000.86360.50000
8310215211671000.06060.42420.42420.04350.06060.42420.01380.13640.50001
8410221651311700.06060.42420.42420.03910.06060.42428.9042e-050.13640.50000
8510221641412600.06060.42420.42420.03910.06060.42421.9630e-040.13640.50000
861021166119800.06060.42420.42420.03910.06060.42422.7199e-050.13640.50000
8710231651412710.06060.42420.42420.03910.06060.42421.3221e-040.13640.50000
8810216213561200.06060.42420.42420.05940.06060.42420.04390.13640.50001
8910215314451300.06060.42420.42420.09340.06060.42420.09190.13640.50001
9010231611851410.78790.42420.42420.78780.78790.42421.00000.86360.50000
9110231541412600.06060.42420.42420.03910.06060.42421.9630e-040.13640.50000
9210231661311800.06060.42420.42420.03910.06060.42425.9968e-050.13640.50000
9310216391014510.78790.42420.42420.78780.78790.42421.00000.86360.50001
9410215212561100.06060.42420.42420.05140.06060.42420.03000.13640.50001
951021159741210.78790.42420.42420.78780.78790.42421.00000.86360.50000
961021517913310.78790.42420.42420.78780.78790.42421.00000.86360.50001
9710221613531510.78790.42420.42420.78770.78790.42420.99960.86360.50000
9810221531412500.06060.42420.42420.03910.06060.42422.9146e-040.13640.50000
9910216515671400.06060.42420.42420.07250.06060.42420.06380.13640.50001
10010215312671100.06060.42420.42420.04650.06060.42420.02040.13640.50001
⋮
ric_model = fitglme(altTrialData, 'Chose1 ~ ad_Prob1 + a0greaterThanb0 + (1 | SubjectID)', 'Distribution', 'binomial');
ric_model.Rsquared
ans = struct with fields:
    Ordinary: 0.6858
    Adjusted: 0.6856

	SubjectID	a0	b0	a1	b1	a2	b2	Chose1	alphaOnly_Prob1	deltaOnly_Prob1	rhoOnly_Prob1	ad_Prob1	ar_Prob1	dr_Prob1	noEpsilon_Prob1	noGamma_Prob1	gammaOnly_Prob1	a0greaterThanb0
1	101	16	5	14	7	8	13	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
2	101	16	1	5	12	15	2	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
3	101	16	3	15	4	5	14	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
4	101	5	16	11	10	7	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
5	101	15	2	8	9	13	4	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
6	101	5	16	7	14	12	9	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
7	101	1	15	12	4	2	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
8	101	2	15	10	7	4	13	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
9	101	15	3	8	10	14	4	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
10	101	3	15	10	8	5	13	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
11	101	15	1	6	10	14	2	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
12	101	3	15	11	7	4	14	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
13	101	15	3	11	7	8	10	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
14	101	14	2	13	3	4	12	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
15	101	15	2	7	10	14	3	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
16	101	3	16	12	7	4	15	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
17	101	15	2	11	6	7	10	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
18	101	2	16	5	13	11	7	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
19	101	2	16	4	14	12	6	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
20	101	1	16	6	11	9	8	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
21	101	3	16	5	14	12	7	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
22	101	16	2	13	5	6	12	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
23	101	15	3	14	4	5	13	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
24	101	3	16	11	8	5	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
25	101	3	15	4	14	12	6	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
26	101	3	16	6	13	11	8	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
27	101	16	3	9	10	14	5	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
28	101	15	2	12	5	6	11	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
29	101	1	15	9	7	4	12	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
30	101	15	1	7	9	13	3	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
31	101	2	16	13	5	3	15	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
32	101	2	15	3	14	12	5	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
33	101	16	5	15	6	7	14	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
34	101	15	3	12	6	7	11	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
35	101	16	2	7	11	15	3	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
36	101	5	16	6	15	13	8	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
37	101	15	1	14	2	3	13	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
38	101	13	2	7	8	12	3	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
39	101	4	16	7	13	11	9	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
40	101	2	16	11	7	4	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
41	101	16	2	11	7	8	10	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
42	101	1	15	11	5	2	14	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
43	101	16	3	8	11	15	4	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
44	101	3	16	10	9	6	13	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
45	101	2	14	9	7	4	12	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
46	101	2	14	10	6	3	13	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
47	101	16	2	15	3	4	14	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
48	101	3	16	4	15	13	6	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
49	101	5	16	12	9	6	15	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
50	101	3	15	5	13	11	7	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
51	101	16	5	10	11	15	6	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
52	101	16	1	7	10	14	3	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
53	101	2	15	4	13	11	6	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
54	101	1	16	13	4	2	15	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
55	101	16	2	8	10	14	4	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
56	101	2	16	12	6	3	15	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
57	101	15	3	13	5	6	12	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
58	101	2	16	6	12	10	8	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
59	101	16	2	12	6	7	11	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
60	101	16	5	13	8	9	12	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
61	101	15	2	14	3	4	13	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
62	101	16	1	6	11	15	2	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
63	101	14	2	7	9	13	3	0	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	1
64	101	1	15	3	13	11	5	1	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	0
65	101	15	2	13	4	5	12	0	0.4545	0.5152	0.5152	0.5152	0.4545	0.5152	0.5606	0.5606	0.5000	1
66	101	2	15	11	6	3	14	1	0.4545	0.3939	0.3939	0.3939	0.4545	0.3939	0.4394	0.4394	0.5000	0
67	102	16	5	14	7	8	13	0	0.0606	0.4242	0.4242	0.0514	0.0606	0.4242	0.0300	0.1364	0.5000	1
68	102	16	1	5	12	15	2	0	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	1
69	102	16	3	15	4	5	14	1	0.0606	0.4242	0.4242	0.1258	0.0606	0.4242	0.1306	0.1364	0.5000	1
70	102	5	16	11	10	7	14	0	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	0
71	102	15	2	8	9	13	4	0	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	1
72	102	5	16	7	14	12	9	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	5.9968e-05	0.1364	0.5000	0
73	102	1	15	12	4	2	14	1	0.7879	0.4242	0.4242	0.7877	0.7879	0.4242	0.9996	0.8636	0.5000	0
74	102	2	15	10	7	4	13	0	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	0
75	102	15	3	8	10	14	4	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	1
76	102	3	15	10	8	5	13	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	0
77	102	15	1	6	10	14	2	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	1
78	102	3	15	11	7	4	14	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	0
79	102	15	3	11	7	8	10	0	0.0606	0.4242	0.4242	0.0417	0.0606	0.4242	0.0094	0.1364	0.5000	1
80	102	14	2	13	3	4	12	0	0.0606	0.4242	0.4242	0.0934	0.0606	0.4242	0.0919	0.1364	0.5000	1
81	102	15	2	7	10	14	3	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	1
82	102	3	16	12	7	4	15	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	0
83	102	15	2	11	6	7	10	0	0.0606	0.4242	0.4242	0.0435	0.0606	0.4242	0.0138	0.1364	0.5000	1
84	102	2	16	5	13	11	7	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	8.9042e-05	0.1364	0.5000	0
85	102	2	16	4	14	12	6	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	1.9630e-04	0.1364	0.5000	0
86	102	1	16	6	11	9	8	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	2.7199e-05	0.1364	0.5000	0
87	102	3	16	5	14	12	7	1	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	1.3221e-04	0.1364	0.5000	0
88	102	16	2	13	5	6	12	0	0.0606	0.4242	0.4242	0.0594	0.0606	0.4242	0.0439	0.1364	0.5000	1
89	102	15	3	14	4	5	13	0	0.0606	0.4242	0.4242	0.0934	0.0606	0.4242	0.0919	0.1364	0.5000	1
90	102	3	16	11	8	5	14	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	0
91	102	3	15	4	14	12	6	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	1.9630e-04	0.1364	0.5000	0
92	102	3	16	6	13	11	8	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	5.9968e-05	0.1364	0.5000	0
93	102	16	3	9	10	14	5	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	1
94	102	15	2	12	5	6	11	0	0.0606	0.4242	0.4242	0.0514	0.0606	0.4242	0.0300	0.1364	0.5000	1
95	102	1	15	9	7	4	12	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	0
96	102	15	1	7	9	13	3	1	0.7879	0.4242	0.4242	0.7878	0.7879	0.4242	1.0000	0.8636	0.5000	1
97	102	2	16	13	5	3	15	1	0.7879	0.4242	0.4242	0.7877	0.7879	0.4242	0.9996	0.8636	0.5000	0
98	102	2	15	3	14	12	5	0	0.0606	0.4242	0.4242	0.0391	0.0606	0.4242	2.9146e-04	0.1364	0.5000	0
99	102	16	5	15	6	7	14	0	0.0606	0.4242	0.4242	0.0725	0.0606	0.4242	0.0638	0.1364	0.5000	1
100	102	15	3	12	6	7	11	0	0.0606	0.4242	0.4242	0.0465	0.0606	0.4242	0.0204	0.1364	0.5000	1
⋮

So there are no convergence warnings here (i.e. that we reached the maximum number of iterations - or attempts to find a solution - without satisfying the tolerance - or the criteria necessary to stop looking). Let's look at the R squared of the standard model for comparison.

model = fitglm(altTrialData, 'Chose1 ~ ad_Prob1', 'Distribution', 'binomial');
model.Rsquared
ans = struct with fields:
          Ordinary: 0.4435
          Adjusted: 0.4434
               LLR: 0.3890
          Deviance: 0.3890
    AdjGeneralized: 0.5542

So there is a lot of unexplained variance in the standard model. Let's compare to the model with only the random intercept now:

ri_model = fitglme(altTrialData, 'Chose1 ~ ad_Prob1 + (1 | SubjectID)', 'Distribution', 'binomial');
ri_model.Rsquared
ans = struct with fields:
    Ordinary: 0.6635
    Adjusted: 0.6635

Wow, so the condition really only explains a very small amount of varance - there's a lot of indivdual variance that the model doesn't pick up. Let's compare this to the 3 norm model:

threeNormmodel = fitglm(trialData, 'Chose1 ~ Prob1', 'Distribution', 'binomial');
threeNormmodel.Rsquared
ans = struct with fields:
          Ordinary: 0.4399
          Adjusted: 0.4398
               LLR: 0.3868
          Deviance: 0.3868
    AdjGeneralized: 0.5518

So somehow there's a lot of indivdual variance that our models just don't detect whatsoever. Let's see if we can trigger a convergence warning by including a random intercept of condition.

ris_model = fitglme(altTrialData, 'Chose1 ~ ad_Prob1 + a0greaterThanb0 + (1 + a0greaterThanb0 | SubjectID)', 'Distribution', 'binomial');
ris_model.Rsquared
ans = struct with fields:
    Ordinary: 0.6891
    Adjusted: 0.6890

No convergence warnings even here. Okay, let's take it onboard that our model is somehow missing 20% of the apparent variance in this behavior. We can proceed to fivefold validation

fivefold = table();
trialData.Prob1_ff = zeros(height(trialData), 1);
for i = 1:length(included_subjects)
    df = grab_data(included_subjects(i));
    df.Prob1 = zeros(height(df), 1);
    order = randperm(height(df));
    A_ff = zeros(1, 5);
    D_ff = zeros(1, 5);  
    B_ff = zeros(1, 5);
    E_ff = zeros(1, 5);
    G_ff = zeros(1, 5);
    df.Pred = zeros(height(df), 1);
 
    for z = 1:5
        j = round((z - 1) * (height(df) / 5) + 1);
        n = round(z * (height(df) / 5));
        withheld = order(j:n);
 
        result_ff = optimize(@of_ad, initial_params([1, 2, 4:6]), lower_bounds([1, 2, 4:6]), upper_bounds([1, 2, 4:6]), df(~ismember(1:height(df), withheld), :));
        
        A_ff(z) = result_ff(1);
        D_ff(z) = result_ff(2);  
        B_ff(z) = result_ff(3);
        E_ff(z) = result_ff(4);
        G_ff(z) = result_ff(5);
 
        pars = [result_ff(1:2), 0, result_ff(3:5)];
        df.Prob1(withheld) = generatePredictions(pars, df(withheld, :));
    end
 
    Deviance_ff = -2 * sum(df.Chose1 .* log(df.Prob1) + (1 - df.Chose1) .* log(1 - df.Prob1));
    fivefold(i, 1:27) = {included_subjects(i), Deviance_ff, ...
        A_ff(1), A_ff(2), A_ff(3), A_ff(4), A_ff(5), ... 
        D_ff(1), D_ff(2), D_ff(3), D_ff(4), D_ff(5), ...
        B_ff(1), B_ff(2), B_ff(3), B_ff(4), B_ff(5), ...
        E_ff(1), E_ff(2), E_ff(3), E_ff(4), E_ff(5), ...
        G_ff(1), G_ff(2), G_ff(3), G_ff(4), G_ff(5)};
    trialData.Prob1_ff(trialData.SubjectID == included_subjects(i)) = df.Prob1;
end
 
fivefold.Properties.VariableNames = {'SubjectID', 'Deviance', ...
    'A_F1', 'A_F2', 'A_F3', 'A_F4', 'A_F5', ...
    'D_F1', 'D_F2', 'D_F3', 'D_F4', 'D_F5', ...
    'B_F1', 'B_F2', 'B_F3', 'B_F4', 'B_F5', ...
    'E_F1', 'E_F2', 'E_F3', 'E_F4', 'E_F5', ...
    'G_F1', 'G_F2', 'G_F3', 'G_F4', 'G_F5'}
fivefold = 57×27 table 
 SubjectIDDevianceA_F1A_F2A_F3A_F4A_F5D_F1D_F2D_F3D_F4D_F5B_F1B_F2B_F3B_F4B_F5E_F1E_F2E_F3E_F4E_F5G_F1G_F2G_F3G_F4G_F5
110197.18800.47820.48110.41451.21030.24821.43071.39621.46190.75691.48696.30716.04205.90124.02854.25330.47640.45000.40380.50000.3964-0.0877-0.0556-0.1190-0.00940.0045
210267.70721.10950.84511.59940.87061.15230.16460.12020.10640.11880.20342.62404.04877.46463.95693.30830.13470.15630.12590.10100.1092-0.5000-0.3297-0.3631-0.2338-0.2933
310321.82851.51471.49241.48811.47351.48820.14520.13560.13680.13300.13556.62146.67896.63056.75606.69550.01864.8216e-050.02090.01860.0200-0.4981-0.0162-0.4984-0.4981-0.4984
410421.93391.59411.59471.47711.63301.59380.10110.10200.13370.07450.09827.49177.50306.67846.32937.55420.01790.01934.8248e-050.02000.01860.49800.4982-0.00890.49830.4981
510522.89190.46124.0252e-063.3869e-060.47916.5183e-061.63341.03241.04261.43770.11885.13740.63970.65386.52235.36810.04000.02190.02980.02281.3567e-05-0.4991-0.5000-0.5000-0.4985-0.3360
610666.94481.61521.55391.61401.60161.59760.10030.09080.10930.09680.09877.63327.53447.50967.43987.60460.18960.11540.15380.19230.1141-0.1836-3.5427e-08-0.3750-0.2000-0.1494
710775.08620.26880.37560.33440.25350.31251.60471.68571.50681.60191.57493.96194.29004.12674.10734.61180.22710.26070.20540.18930.2464-0.0283-0.1164-0.1957-0.0283-0.1243
810893.23990.30161.98680.82200.82591.1661e-040.90491.36040.67500.64621.01940.09939.93821.79471.78910.17500.13230.38780.35500.39170.4661-0.4999-0.1668-0.2839-0.1939-0.1502
910977.75701.74920.36730.27490.20871.2049e-071.70061.30901.66911.53900.05949.35966.14923.87124.48403.72880.44220.32920.34620.32640.2733-0.2234-0.3178-0.2778-0.3298-0.2838
1011026.95988.7831e-063.4212e-064.6848e-065.3349e-060.43141.04701.03471.04261.04281.54890.72700.67220.67440.72694.04120.06350.05380.07440.05160.0417-0.5000-0.5000-0.5000-0.5000-0.4992
111110.00691.50291.49051.48741.49281.49220.12770.13520.13890.13990.13566.13196.66386.68066.66706.68154.7490e-054.7845e-054.8289e-054.7938e-054.7864e-050.0954-0.0182-0.01390.0012-0.0180
1211289.18371.19140.40651.57401.60070.61541.36150.87951.58881.63420.74237.32830.40129.60379.19710.45220.27470.23220.25600.27670.0976-0.0141-0.0816-0.1663-0.1917-0.1630
131130.00681.49061.49051.49031.48901.48870.13950.13510.13890.13620.13396.66586.66426.65196.64266.63814.7844e-054.7751e-054.8447e-054.7911e-054.7794e-05-0.0067-2.9492e-04-0.0252-0.0123-0.0077
1411468.93811.05941.06121.07171.07671.04190.17750.10860.10390.14383.1824e-063.90651.54001.79831.95561.01450.15690.20090.16660.18650.1779-0.3593-0.2982-0.2821-0.2937-0.5000
1511540.61521.49781.48201.48181.47251.47890.13620.13940.13370.13310.13816.56656.48426.62316.55916.56630.13340.16670.11110.13050.1400-0.4997-0.4998-0.4997-0.4998-0.4998
1611631.76681.49661.49571.48991.48561.53330.13650.13140.13930.13700.14106.64366.37176.54566.56456.54030.08340.09260.08000.10000.0962-0.4996-0.4996-0.4996-0.4996-0.4997
1711782.43320.43780.24010.24701.25041.21971.58811.52381.56191.99991.98594.44404.26534.21849.99979.93550.27110.25300.19230.27920.2284-0.0902-0.0765-0.2000-0.01150.0472
1811861.24371.60631.56291.62491.57491.57300.09980.09840.10390.08960.10187.61177.46657.25267.34947.35610.20840.18670.20240.23810.1538-0.3345-0.3929-0.2941-0.3250-0.3750
1911922.15911.51631.49211.48491.48571.49230.13450.13570.15180.13700.13536.45286.70096.61866.87136.67780.02004.7720e-050.01730.01930.0209-0.49840.0011-0.4979-0.4983-0.4985
2012093.52680.51271.45160.86700.52151.28871.48101.99931.11721.46940.80204.00349.99654.10143.99696.07050.50000.47990.50000.50000.5000-0.0192-0.0849-0.0769-0.0962-0.1538
2112188.61960.98550.62431.50580.94081.06270.16091.36830.08001.11760.15220.39743.99356.77584.11891.72400.22010.50000.31550.50000.3120-0.4577-0.1346-0.1604-0.2115-0.3582
221220.00711.47531.49101.48671.48701.49000.13370.13580.13590.14890.13526.74886.69946.63926.66186.66794.7765e-054.7729e-054.8605e-054.7861e-054.7751e-05-0.0094-0.0139-0.02860.0036-0.0066
2312486.45892.00001.04532.00000.98101.59950.57610.16380.36350.19020.09459.99991.61759.99994.21377.35080.30560.25180.29820.30130.2629-0.04560.0600-0.02000.05170.0246
2412523.62351.20721.09571.11331.10131.59300.08420.09450.10560.10770.10373.58741.77172.56222.35587.50270.03944.8474e-060.02920.03280.01860.4991-0.34800.49880.49920.4981
2512673.86460.92761.57601.11121.56361.57431.12160.09990.85600.09340.09544.05697.40954.12427.22487.28280.50000.41670.50000.38770.3681-0.2255-0.3000-0.2885-0.2937-0.3491
2612791.77831.22101.61041.59361.09101.61370.68050.10250.09040.85440.10094.05007.77147.50323.99117.64680.50000.35750.36540.50000.4289-0.0472-0.0697-0.1842-0.1038-0.0628
2712865.55101.49081.50221.49001.49151.47580.13530.13790.14130.13530.13506.66236.70916.73146.68316.54700.05360.05560.06254.7776e-050.0536-0.4994-0.4994-0.4995-0.0197-0.4994
2812991.68511.28441.25991.28520.25351.26132.00002.00002.00001.42572.00009.99999.99999.99994.959810.00000.34590.33040.33620.29260.3889-0.0545-0.1048-0.0743-0.1835-0.0707
2913021.86171.50041.49011.49181.49831.51370.13970.13570.13620.14060.13776.70156.70386.66986.66436.66410.02004.8635e-050.01790.02090.0193-0.4983-0.0050-0.4980-0.4984-0.4982
3013142.62821.61211.59151.56301.59161.60810.10090.09930.08620.10150.10127.60517.50007.50727.39707.57030.09570.09140.09620.09140.0757-0.1269-0.2812-0.1000-0.2812-0.0283
3113259.74710.31821.5791e-050.30310.47480.25811.68841.03121.52231.54281.49094.01370.70174.82756.59694.43050.14500.08160.08000.14890.1339-0.3621-0.3464-0.4996-0.3708-0.3617
3213385.36850.38260.54850.52990.50901.46390.88600.79650.86030.82371.99950.39320.49380.96360.32479.99770.26020.26630.33550.18040.3591-0.2117-0.4033-0.2280-0.5000-0.2153
3313490.55351.57471.06601.03100.69501.04590.08950.18450.20231.28440.13737.34021.88551.53303.99971.37850.32520.37700.30640.50000.3901-0.0695-0.2106-0.1822-0.0769-0.1387
3413593.35051.40441.51991.51931.27321.86122.00001.99881.99772.00001.999610.00009.99419.98899.99989.99820.32040.33610.36520.33170.3571-0.0920-0.1878-0.1346-0.1985-0.2000
3513655.55981.60771.56681.46961.58551.59440.10120.09250.13540.10410.09987.45307.41816.54937.52467.48390.13390.10930.09620.11470.0762-0.3617-0.3170-0.4996-0.3385-0.2570
3613884.03170.24120.45760.23440.26160.23970.69221.42811.57191.46971.49394.42876.68814.03904.53174.19480.44030.33640.31190.34520.3523-0.2380-0.0946-0.3312-0.2241-0.1452
3714096.01680.48210.29431.00510.90871.71321.47491.55331.25051.14562.00006.29524.17706.26995.454510.00000.42560.40480.34220.30980.32010.03850.01470.0179-0.0233-0.0371
3814191.34651.59591.55551.51700.62242.00001.60821.55381.51641.17301.75109.03828.81079.03244.192410.00000.36680.46880.40050.50000.4027-0.1632-0.0511-0.1072-0.0686-0.0776
3914298.55301.58101.56591.53440.69301.99870.09330.09430.08931.28391.40047.53247.38127.23233.98679.99340.39670.41450.41370.50000.4116-0.0332-0.0361-0.1475-0.0283-0.2004
4014386.40860.47761.59691.53961.01481.62141.51360.09350.08870.95400.09063.98627.45917.51234.12217.49840.50000.26920.24670.50000.2197-0.1154-0.2857-0.1757-0.1346-0.1207
4114421.51131.48381.48581.49741.49281.49210.13720.13930.13440.13410.13596.63456.61166.60866.61586.64940.01930.01790.02090.02094.9496e-05-0.4982-0.4980-0.4984-0.4984-0.0319
4214589.79920.62791.54072.00001.99971.82721.20211.50860.60971.85111.76565.31028.58529.99989.99879.52030.50000.39180.33650.40720.4254-0.1400-0.1666-0.2654-0.2368-0.2103
4314693.63280.57420.34592.00002.00002.00001.41951.09641.54841.65181.64864.01282.94939.99999.99999.99990.50000.47930.40070.40780.4269-0.1415-0.2079-0.1872-0.2242-0.2028
441470.00730.45150.46630.56520.47830.63121.36641.35791.33561.34471.46654.19474.11235.81324.14455.33484.8934e-055.6209e-055.4811e-054.8731e-055.4675e-05-0.00550.0365-0.00479.4005e-040.0123
4514834.11811.49201.50591.48291.55871.47450.13820.13370.13430.14930.13016.70436.53766.53766.63356.55640.08340.06000.11540.09680.0962-0.4996-0.4995-0.4997-0.4996-0.4997
4614922.69402.00001.99971.99971.99951.48980.38120.36350.36350.36350.13889.99999.99869.99869.99736.66153.2270e-063.0099e-052.9809e-054.2908e-054.7838e-05-0.0168-0.3240-0.3212-0.27590.0060
4715080.66910.57980.52300.12230.31171.40460.99000.84800.99451.40752.00002.85083.91320.15224.05519.99990.29890.28810.07310.33590.3198-0.1814-0.2926-0.5000-0.1617-0.2010
4815163.02351.58171.56931.63901.58801.58850.08510.09470.09030.09320.09877.32717.55877.37677.52007.52510.21970.23190.23520.20820.2121-0.1207-0.2187-0.0951-0.1725-0.2857
4915267.99101.63191.02511.05261.09161.05770.09070.01390.07381.6190e-064.8798e-067.40820.83381.18821.08741.05270.18870.14510.14350.21150.1896-0.0094-0.1382-0.1618-0.1102-0.0828
50153101.90170.29990.34750.35070.37301.83611.56101.62431.44131.59281.99854.43824.00624.14184.65159.99240.37680.41310.44220.35210.4870-0.04060.12760.00250.15920.0436
5115442.29131.59751.50541.60391.59781.60220.09530.13510.10610.10210.10007.34756.61707.53077.50057.58030.03770.01860.03870.01790.03810.0094-0.4982-0.03850.49800.0472
5215549.11392.00002.00002.00001.59112.00000.37880.36270.36240.10230.361810.000010.000010.00007.48509.99980.03370.05730.06030.03770.04140.05040.13730.1820-0.00940.4994
5315669.60660.49970.49800.48440.47000.45420.96350.96250.94791.44340.92382.22762.22721.89836.55701.70220.04110.03820.03610.03710.0019-0.3919-0.3501-0.4648-0.49910.4999
54157116.69871.01252.00001.99101.99832.00001.9405e-060.75021.34761.37370.75080.286810.00009.96029.991610.00000.16000.28690.26470.23000.3318-0.5000-0.1365-0.3158-0.5000-0.1669
5515892.96590.64041.27780.25271.24541.29041.20542.00001.53010.78412.00004.09109.99984.11913.98349.99990.50000.44620.43750.50000.4643-0.0660-0.0789-0.0714-0.0660-0.0572
5616059.19460.49230.31680.49570.26250.34831.45531.55041.44991.38841.55046.70614.70916.36445.01014.81790.15560.17310.17110.11560.15260.14290.05560.1494-0.01920.1068
5716190.52720.48901.30001.41860.33440.25931.42120.42090.42461.60681.54536.41444.09704.07264.56084.22880.38460.50000.50000.37200.3958-0.2000-0.1923-0.2115-0.2200-0.1316

Now we can check the model accuracy

sum(trialData.Chose1 == round(trialData.Prob1_ff))/height(trialData)
ans = 0.7681

That's not such a bad drop in model accuracy - only around 2%. Now let's test it agains the normally recovered model:

fivefold.BIC = fivefold.Deviance + log(65) * 6;
[~, p, ~, stats] = ttest(fivefold.BIC, altSubjectData.BIC_M4); stats.p = round(p, 3)
stats = struct with fields:
    tstat: 11.0559
       df: 56
       sd: 9.8826
        p: 0

So it's significantly better, but that's not surprising and not concerning given the small decrease in accuracy. Let's look at the similarlity of the parameters - first, let's compute cosine similarity.

cosines = zeros(1, 25);
 
function cos = cosine(A, B)
cos = dot(A, B) / (norm(A) * norm(B));
end
 
for i = 1:5
    cosines(i) = cosine(altSubjectData.Alpha_M4, fivefold{:, i + 2});
    cosines(i + 5) = cosine(altSubjectData.Delta_M4, fivefold{:, i + 7});
    cosines(i + 10) = cosine(altSubjectData.Beta_M4, fivefold{:, i + 12});
    cosines(i + 15) = cosine(altSubjectData.Epsilon_M4, fivefold{:, i + 17});
    cosines(i + 20) = cosine(altSubjectData.Gamma_M4, fivefold{:, i + 22});
end

Now let's see the cosine for all parameters - first alpha:

mean(cosines(1:5))
ans = 0.9489

Very good. Now delta

mean(cosines(6:11))
ans = 0.9336

Again, very good. Now beta

mean(cosines(11:15))
ans = 0.9235

Also good. Now epsilon

mean(cosines(16:21))
ans = 0.9777

Nice. Finally gamma

mean(cosines(21:25))
ans = 0.8917

Everything looks good here. Let's go to the fun part - model testing.

3.1 Compare Models

First, let's test the alpha + delta model agains the full model (which has alpha + delta + rho)

[~, p, ~, stats] = ttest(altSubjectData.BIC_M4, subjectData.BIC); stats.p = round(p, 3)
stats = struct with fields:
    tstat: -13.4929
       df: 56
       sd: 2.4684
        p: 0

So M4 has a significantly lower BIC that the full model (i.e. p < 0.001), meaning it is significantly better to ignore rank-reversal. How about the alpha only model (M1)

[~, p, ~, stats] = ttest(altSubjectData.BIC_M4, altSubjectData.BIC_M1); stats.p = round(p, 3)
stats = struct with fields:
    tstat: -1.5722
       df: 56
       sd: 20.1206
        p: 0.1220

So M4 is better, but not significantly better meaning that including harm aversion does not significanlty improve model performance. How about the delta only model (M2)

[~, p, ~, stats] = ttest(altSubjectData.BIC_M4, altSubjectData.BIC_M2); stats.p = round(p, 3)
stats = struct with fields:
    tstat: -5.2137
       df: 56
       sd: 33.6938
        p: 0

So including inequality aversion significantly improves model performance. How about the rho only model (M3)

[~, p, ~, stats] = ttest(altSubjectData.BIC_M4, altSubjectData.BIC_M3); stats.p = round(p, 3)
stats = struct with fields:
    tstat: -5.2403
       df: 56
       sd: 33.6367
        p: 0

Including both inequality aversion and harm aversion significantly improves model performance. Let's look at the inequality and rank reversal aversion model (M5)

[~, p, ~, stats] = ttest(altSubjectData.BIC_M4, altSubjectData.BIC_M5); stats.p = round(p, 3)
stats = struct with fields:
    tstat: -1.7419
       df: 56
       sd: 2.3230
        p: 0.0870

Switching harm aversion for rank reversal aversion does not significanlty lessen model performance. Interesting - now we can examine the harm and rank reversal aversion model (M6)

[~, p, ~, stats] = ttest(altSubjectData.BIC_M4, altSubjectData.BIC_M6); stats.p = round(p, 3)
stats = struct with fields:
    tstat: -6.1536
       df: 56
       sd: 33.6839
        p: 0

Switching inequality aversion for rank reversal aversion significantly lessens model performance. How about if we remove the noise and bias parameters from the full model

[~, p, ~, stats] = ttest(altSubjectData.BIC_M4, altSubjectData.BIC_M7); stats.p = round(p, 3)
stats = struct with fields:
    tstat: -3.5271
       df: 56
       sd: 100.1979
        p: 1.0000e-03

So it does significantly hurt the model performance compared to our favored model - but let's compare apples to apples here:

[~, p, ~, stats] = ttest(subjectData.BIC, altSubjectData.BIC_M7); stats.p = round(p, 3)
stats = struct with fields:
    tstat: -3.1910
       df: 56
       sd: 100.3137
        p: 0.0020

So including these significantly improves model performance. We may wish to verify by instantiating this model as a variant of M4 instead of the full model, but at the moment it's not extremely urgent to do. Let's move onto M8 which only removes the bias parameter

[~, p, ~, stats] = ttest(subjectData.BIC, altSubjectData.BIC_M8); stats.p = round(p, 3)
stats = struct with fields:
    tstat: 1.3248
       df: 56
       sd: 3.8013
        p: 0.1910

It seems that including the bias parameter might not be extremely important, so we again may wish to rerun M8 as a variant of M4. But that is for another time. Let's finish up with M9 and that only has left right preferences

[~, p, ~, stats] = ttest(altSubjectData.BIC_M4, altSubjectData.BIC_M9); stats.p = round(p, 3); stats
stats = struct with fields:
    tstat: -4.9360
       df: 56
       sd: 31.8217
        p: 0

So that's significant, but not as big as we might expect.

Here, we can conclude that inequality aversion plays a significant role in these decisions, but harm aversion does not. Also, these decisions tend to be very noisy, but not significantly determined by biases for chosing left versus right.

3.3 Test for Individual Differences

Let’s first recover parameters over the whole dataset and we can assess how accurate it is

resultNID = optimize(@of_ad, initial_params([1,2,4:6]), lower_bounds([1,2,4:6]), upper_bounds([1,2,4:6]), trialData);
pars = [resultNID(1:2), 0, resultNID(3:5)];
trialData.Prob1_NID = generatePredictions(pars, trialData);
sum(trialData.Chose1 == round(trialData.Prob1_NID))/height(trialData)
ans = 0.5967

Not very good. This isn’t surprising given that people often have very different preferences. Now let’s test for individual differences.

altSubjectData.Deviance_NID = zeros(size(altSubjectData,1), 1);
 
for i = 1:length(included_subjects)
    trials = find(included_subjects(i) == trialData.SubjectID);
    df = trialData(trials, :);
    altSubjectData.Deviance_NID(i) = -2 * sum(df.Chose1 .* log(df.Prob1_NID) + (1 - df.Chose1) .* log(1 - df.Prob1_NID));
end
 
altSubjectData.BIC_NID = altSubjectData.Deviance_NID + log(65) * 5 / length(included_subjects);
[~,p,~,stats] = ttest(altSubjectData.BIC_M4, altSubjectData.BIC_NID); stats.p = round(p, 3); stats
stats = struct with fields:
    tstat: -3.6358
       df: 56
       sd: 30.0943
        p: 1.0000e-03

Significant individual differences. Let’s see which models are worse

find(modelBIC > sum(altSubjectData.BIC_NID))
ans = 1×5
     3     4     7     8    10

So M2 (delta only), M3 (rho only), M6 (delta + rho), M7 (no epsilon or gamma), and M9 (only left/right) are worse than this model

	a0	b0	a1	b1	a2	b2
1	18	2	7	13	14	6
2	19	1	16	4	10	10
3	11	9	7	13	10	10
4	20	0	13	7	10	10
5	16	4	6	14	9	11
6	11	9	9	11	11	9
7	13	7	7	13	12	8
8	16	4	12	8	11	9
9	20	0	16	4	14	6
10	20	0	16	4	14	6
11	11	9	8	12	6	14
12	20	0	6	14	9	11
13	20	0	8	12	12	8
14	15	5	15	5	7	13
⋮

	subject	trial_num	run	partnerA	partnerB	initial_A	initial_B	Pie_initial_high	Pie_initial_low	diff_initial	transfer_size_1	fairness_1	A_final_1	B_final_1	Pie_Alter1_ini_high	Pie_Alter1_ini_low	diff_final_1	reverse_1	transfer_size_2	fairness_2	A_final_2	B_final_2	Pie_Alter2_ini_high	Pie_Alter2_ini_low	Equal_alter_pie_ini_high	Equal_alter_pie_ini_low	Unqual_alter_pie_ini_high	Unequal_alter_pie_ini_low	diff_final_2	reverse_2	transfer_diff	fairness_diff	trail_type	order_option	order_orientation	HPP_key	reaction	If_Rev	If_Rev_1	Revsize1	Revsize2	TransferSize1_a	TransferSize2_a	TransferSize1_b	TransferSize2_b	Mah	Mal	Mbh	Mbl	Trans1_Diff	Trans2_Diff
1	101	1	1	46	43	15	2	15	2	13	5	10	10	7	10	7	3	0	1	2	14	3	14	3	10	7	14	3	11	0	4	8	1	2	1	1	0	0	0	0	0	5	0	1	0	10	7	14	3	4	0
2	101	2	1	82	1	16	5	16	5	11	2	4	14	7	14	7	7	0	8	6	8	13	8	13	8	13	14	7	5	1	6	2	3	1	1	2	0	1	1	0	5	2	0	5.5000	2.5000	14	7	8	13	3.5000	2.5000
3	101	3	1	2	3	16	5	16	5	11	2	4	14	7	14	7	7	0	3	6	13	8	13	8	13	8	14	7	5	0	1	2	1	1	1	2	0	0	0	0	0	2	0	3	0	14	7	13	8	1	0
4	101	4	1	42	47	15	3	15	3	12	5	10	10	8	10	8	2	0	1	2	14	4	14	4	10	8	14	4	10	0	4	8	1	2	1	1	0	0	0	0	0	5	0	1	0	10	8	14	4	4	0
5	101	5	1	62	67	5	15	15	5	10	7	6	12	8	8	12	4	1	5	10	10	10	10	10	10	10	8	12	0	1	-2	4	5	1	2	2	0	1	1	4	0	5	2	5	0	8	12	10	10	0	2
6	101	6	1	60	23	1	15	15	1	14	3	6	4	12	12	4	8	0	1	2	2	14	14	2	12	4	14	2	12	0	2	4	1	2	2	1	0	0	0	0	0	3	0	1	0	12	4	14	2	2	0
7	101	7	1	44	47	16	1	16	1	15	11	8	5	12	5	12	7	1	1	2	15	2	15	2	5	12	15	2	13	0	10	6	3	2	1	1	0	1	1	7	0	7.5000	3.5000	1	0	5	12	15	2	6.5000	3.5000
8	101	8	1	24	29	2	16	16	2	14	3	6	5	13	13	5	8	0	5	10	7	11	11	7	11	7	13	5	4	0	2	4	1	1	2	2	0	0	0	0	0	3	0	5	0	13	5	11	7	2	0
9	101	9	1	20	45	16	3	16	3	13	1	2	15	4	15	4	11	0	11	4	5	14	5	14	5	14	15	4	9	1	10	2	3	1	1	1	1	1	1	0	9	1	0	6.5000	4.5000	15	4	5	14	5.5000	4.5000
10	101	10	1	16	53	15	2	15	2	13	6	12	9	8	9	8	1	0	2	4	13	4	13	4	9	8	13	4	9	0	4	8	1	2	1	1	0	0	0	0	0	6	0	2	0	9	8	13	4	4	0
11	101	11	1	20	45	5	15	15	5	10	3	6	8	12	12	8	4	0	5	10	10	10	10	10	10	10	12	8	0	1	2	4	4	1	2	2	0	1	1	0	0	3	0	5	0	12	8	10	10	2	0
12	101	12	1	32	21	1	16	16	1	15	5	10	6	11	11	6	5	0	7	14	8	9	9	8	9	8	11	6	1	0	2	4	1	1	2	1	1	0	0	0	0	5	0	7	0	11	6	9	8	2	0
13	101	13	1	22	23	5	16	16	5	11	6	10	11	10	10	11	1	1	2	4	7	14	14	7	10	11	14	7	7	0	4	6	3	2	2	1	0	1	1	1	0	5.5000	0.5000	2	0	10	11	14	7	3.5000	0.5000
14	101	14	1	70	69	2	16	16	2	14	2	4	4	14	14	4	10	0	4	8	6	12	12	6	12	6	14	4	6	0	2	4	1	1	2	2	0	0	0	0	0	2	0	4	0	14	4	12	6	2	0
15	101	15	1	18	23	15	2	15	2	13	7	12	8	9	8	9	1	1	2	4	13	4	13	4	8	9	13	4	9	0	5	8	3	2	1	1	0	1	1	1	0	6.5000	0.5000	2	0	8	9	13	4	4.5000	0.5000
16	101	16	1	56	29	3	16	16	3	13	4	8	7	12	12	7	5	0	1	2	4	15	15	4	12	7	15	4	11	0	3	6	1	2	2	1	0	0	0	0	0	4	0	1	0	12	7	15	4	3	0
17	101	17	1	34	39	5	16	16	5	11	2	4	7	14	14	7	7	0	7	8	12	9	9	12	9	12	14	7	3	1	5	4	3	1	2	2	0	1	1	0	0	2	0	5.5000	1.5000	14	7	9	12	3.5000	1.5000
18	101	18	1	38	9	1	15	15	1	14	11	6	12	4	4	12	8	1	1	2	2	14	14	2	4	12	14	2	12	0	10	4	3	2	2	1	0	1	1	8	0	7	4	1	0	4	12	14	2	6	4
19	101	19	1	62	21	2	15	15	2	13	8	10	10	7	7	10	3	1	2	4	4	13	13	4	7	10	13	4	9	0	6	6	3	2	2	1	0	1	1	3	0	6.5000	1.5000	2	0	7	10	13	4	4.5000	1.5000
20	101	20	1	82	1	3	15	15	3	12	6	12	9	9	9	9	0	1	9	6	12	6	6	12	9	9	6	12	6	1	-3	6	5	2	2	1	0	1	1	0	6	6	0	6	3	9	9	6	12	0	3
21	101	21	1	8	69	2	14	14	2	12	7	10	9	7	7	9	2	1	6	12	8	8	8	8	8	8	7	9	0	1	-1	2	5	1	2	2	0	1	1	2	0	6	1	6	0	7	9	8	8	0	1
22	101	22	1	30	31	3	16	16	3	13	2	4	5	14	14	5	9	0	4	8	7	12	12	7	12	7	14	5	5	0	2	4	1	1	2	2	0	0	0	0	0	2	0	4	0	14	5	12	7	2	0
23	101	23	1	40	5	15	3	15	3	12	7	10	8	10	8	10	2	1	1	2	14	4	14	4	8	10	14	4	10	0	6	8	3	2	1	1	0	1	1	2	0	6	1	1	0	8	10	14	4	5	1
24	101	24	1	52	35	3	15	15	3	12	7	10	10	8	8	10	2	1	2	4	5	13	13	5	8	10	13	5	8	0	5	6	3	2	2	1	0	1	1	2	0	6	1	2	0	8	10	13	5	4	1
25	101	25	1	66	65	2	15	15	2	13	5	10	7	10	10	7	3	0	2	4	4	13	13	4	10	7	13	4	9	0	3	6	1	2	2	1	0	0	0	0	0	5	0	2	0	10	7	13	4	3	0
26	101	26	1	50	37	15	1	15	1	14	9	10	6	10	6	10	4	1	1	2	14	2	14	2	6	10	14	2	12	0	8	8	3	2	1	1	0	1	1	4	0	7	2	1	0	6	10	14	2	6	2
27	101	27	1	50	37	16	3	16	3	13	1	2	15	4	15	4	11	0	2	4	14	5	14	5	14	5	15	4	9	0	1	2	1	1	1	2	0	0	0	0	0	1	0	2	0	15	4	14	5	1	0
28	101	28	1	38	9	3	15	15	3	12	8	8	11	7	7	11	4	1	1	2	4	14	14	4	7	11	14	4	10	0	7	6	3	2	2	2	1	1	1	4	0	6	2	1	0	7	11	14	4	5	2
29	101	29	1	46	47	3	15	15	3	12	5	10	8	10	10	8	2	0	2	4	5	13	13	5	10	8	13	5	8	0	3	6	1	2	2	1	0	0	0	0	0	5	0	2	0	10	8	13	5	3	0
30	101	30	1	24	37	16	2	16	2	14	6	12	10	8	10	8	2	0	7	14	9	9	9	9	9	9	10	8	0	1	1	2	4	1	1	2	0	1	1	0	0	6	0	7	0	10	8	9	9	1	0
31	101	31	1	30	25	2	14	14	2	12	5	10	7	9	9	7	2	0	6	12	8	8	8	8	8	8	9	7	0	1	1	2	4	1	2	1	1	1	1	0	0	5	0	6	0	9	7	8	8	1	0
32	101	32	1	72	71	15	3	15	3	12	4	8	11	7	11	7	4	0	7	10	8	10	8	10	8	10	11	7	2	1	3	2	3	1	1	1	1	1	1	0	2	4	0	6	1	11	7	8	10	2	1
33	101	33	1	54	33	15	1	15	1	14	5	10	10	6	10	6	4	0	1	2	14	2	14	2	10	6	14	2	12	0	4	8	1	2	1	2	1	0	0	0	0	5	0	1	0	10	6	14	2	4	0
34	101	34	1	40	45	14	2	14	2	12	6	12	8	8	8	8	0	1	9	6	5	11	5	11	8	8	5	11	6	1	-3	6	5	2	1	2	1	1	1	0	6	6	0	6	3	8	8	5	11	0	3
35	101	35	1	6	73	5	16	16	5	11	5	10	10	11	11	10	1	0	2	4	7	14	14	7	11	10	14	7	7	0	3	6	1	2	2	1	0	0	0	0	0	5	0	2	0	11	10	14	7	3	0
36	101	36	1	32	37	14	2	14	2	12	1	2	13	3	13	3	10	0	10	4	4	12	4	12	4	12	13	3	8	1	9	2	3	1	1	1	1	1	1	0	8	1	0	6	4	13	3	4	12	5	4
37	101	37	1	22	41	15	2	15	2	13	8	10	7	10	7	10	3	1	1	2	14	3	14	3	7	10	14	3	11	0	7	8	3	2	1	1	0	1	1	3	0	6.5000	1.5000	1	0	7	10	14	3	5.5000	1.5000
38	101	38	1	54	33	14	2	14	2	12	1	2	13	3	13	3	10	0	2	4	12	4	12	4	12	4	13	3	8	0	1	2	1	1	1	2	0	0	0	0	0	1	0	2	0	13	3	12	4	1	0
39	101	39	1	20	21	3	16	16	3	13	9	8	12	7	7	12	5	1	1	2	4	15	15	4	7	12	15	4	11	0	8	6	3	2	2	1	0	1	1	5	0	6.5000	2.5000	1	0	7	12	15	4	5.5000	2.5000
40	101	40	1	12	61	15	2	15	2	13	4	8	11	6	11	6	5	0	8	10	7	10	7	10	7	10	11	6	3	1	4	2	3	1	1	1	1	1	1	0	3	4	0	6.5000	1.5000	11	6	7	10	2.5000	1.5000
41	101	41	1	24	37	14	2	14	2	12	6	12	8	8	8	8	0	1	3	6	11	5	11	5	8	8	11	5	6	0	3	6	4	2	1	2	1	1	1	0	0	6	0	3	0	8	8	11	5	3	0
42	101	42	1	58	63	16	2	16	2	14	3	6	13	5	13	5	8	0	4	8	12	6	12	6	12	6	13	5	6	0	1	2	1	1	1	2	0	0	0	0	0	3	0	4	0	13	5	12	6	1	0
43	101	43	1	6	73	15	2	15	2	13	4	8	11	6	11	6	5	0	5	10	10	7	10	7	10	7	11	6	3	0	1	2	1	1	1	2	0	0	0	0	0	4	0	5	0	11	6	10	7	1	0
44	101	44	1	12	13	16	1	16	1	15	4	8	12	5	12	5	7	0	1	2	15	2	15	2	12	5	15	2	13	0	3	6	1	2	1	1	0	0	0	0	0	4	0	1	0	12	5	15	2	3	0
45	101	45	1	3	11	2	16	16	2	14	3	6	5	13	13	5	8	0	9	10	11	7	7	11	7	11	13	5	4	1	6	4	3	1	2	1	1	1	1	0	0	3	0	7	2	13	5	7	11	4	2
46	101	46	1	42	53	2	16	16	2	14	2	4	4	14	14	4	10	0	10	8	12	6	6	12	6	12	14	4	6	1	8	4	3	1	2	2	0	1	1	0	0	2	0	7	3	14	4	6	12	5	3
47	101	47	1	30	25	15	3	15	3	12	4	8	11	7	11	7	4	0	5	10	10	8	10	8	10	8	11	7	2	0	1	2	1	1	1	2	0	0	0	0	0	4	0	5	0	11	7	10	8	1	0
48	101	48	1	58	27	3	15	15	3	12	6	12	9	9	9	9	0	1	3	6	6	12	12	6	9	9	12	6	6	0	3	6	4	2	2	1	0	1	1	0	0	6	0	3	0	9	9	12	6	3	0
49	101	49	1	52	57	1	16	16	1	15	5	10	6	11	11	6	5	0	8	14	9	8	8	9	8	9	11	6	1	1	3	4	3	1	2	2	0	1	1	0	0	5	0	7.5000	0.5000	11	6	8	9	2.5000	0.5000
50	101	50	1	48	41	3	16	16	3	13	2	4	5	14	14	5	9	0	9	8	12	7	7	12	7	12	14	5	5	1	7	4	3	1	2	2	0	1	1	0	0	2	0	6.5000	2.5000	14	5	7	12	4.5000	2.5000
51	101	51	1	46	43	16	2	16	2	14	3	6	13	5	13	5	8	0	10	8	6	12	6	12	6	12	13	5	6	1	7	2	3	1	1	1	1	1	1	0	6	3	0	7	3	13	5	6	12	4	3
52	101	52	1	14	57	16	2	16	2	14	8	12	8	10	8	10	2	1	7	14	9	9	9	9	9	9	8	10	0	1	-1	2	5	1	1	2	0	1	1	2	0	7	1	7	0	8	10	9	9	0	1
53	101	53	1	42	43	3	15	15	3	12	4	8	7	11	11	7	4	0	1	2	4	14	14	4	11	7	14	4	10	0	3	6	1	2	2	1	0	0	0	0	0	4	0	1	0	11	7	14	4	3	0
54	101	54	1	28	33	5	16	16	5	11	2	4	7	14	14	7	7	0	4	8	9	12	12	9	12	9	14	7	3	0	2	4	1	1	2	2	0	0	0	0	0	2	0	4	0	14	7	12	9	2	0
55	101	55	2	36	13	13	2	13	2	11	5	10	8	7	8	7	1	0	1	2	12	3	12	3	8	7	12	3	9	0	4	8	1	2	1	1	0	0	0	0	0	5	0	1	0	8	7	12	3	4	0
56	101	56	2	42	53	15	3	15	3	12	1	2	14	4	14	4	10	0	10	4	5	13	5	13	5	13	14	4	8	1	9	2	3	1	1	2	0	1	1	0	8	1	0	6	4	14	4	5	13	5	4
57	101	57	2	34	17	3	16	16	3	13	8	10	11	8	8	11	3	1	2	4	5	14	14	5	8	11	14	5	9	0	6	6	3	2	2	1	0	1	1	3	0	6.5000	1.5000	2	0	8	11	14	5	4.5000	1.5000
58	101	58	2	30	35	15	3	15	3	12	3	6	12	6	12	6	6	0	4	8	11	7	11	7	11	7	12	6	4	0	1	2	1	1	1	1	1	0	0	0	0	3	0	4	0	12	6	11	7	1	0
59	101	59	2	66	7	2	15	15	2	13	1	2	3	14	14	3	11	0	3	6	5	12	12	5	12	5	14	3	7	0	2	4	1	1	2	2	0	0	0	0	0	1	0	3	0	14	3	12	5	2	0
60	101	60	2	48	47	2	16	16	2	14	9	10	11	7	7	11	4	1	7	14	9	9	9	9	9	9	7	11	0	1	-2	4	5	1	2	2	0	1	1	4	0	7	2	7	0	7	11	9	9	0	2
61	101	61	2	22	27	3	15	15	3	12	1	2	4	14	14	4	10	0	9	6	12	6	6	12	6	12	14	4	6	1	8	4	3	1	2	2	0	1	1	0	0	1	0	6	3	14	4	6	12	5	3
62	101	62	2	7	31	3	16	16	3	13	3	6	6	13	13	6	7	0	8	10	11	8	8	11	8	11	13	6	3	1	5	4	3	1	2	2	0	1	1	0	0	3	0	6.5000	1.5000	13	6	8	11	3.5000	1.5000
63	101	63	2	16	53	16	3	16	3	13	7	12	9	10	9	10	1	1	2	4	14	5	14	5	9	10	14	5	9	0	5	8	3	2	1	1	0	1	1	1	0	6.5000	0.5000	2	0	9	10	14	5	4.5000	0.5000
64	101	64	2	50	49	15	2	15	2	13	3	6	12	5	12	5	7	0	9	8	6	11	6	11	6	11	12	5	5	1	6	2	3	1	1	2	0	1	1	0	5	3	0	6.5000	2.5000	12	5	6	11	3.5000	2.5000
65	101	65	2	26	33	1	15	15	1	14	8	12	9	7	7	9	2	1	3	6	4	12	12	4	7	9	12	4	8	0	5	6	3	2	2	1	0	1	1	2	0	7	1	3	0	7	9	12	4	4	1
66	101	66	2	56	61	15	1	15	1	14	7	14	8	8	8	8	0	1	10	8	5	11	5	11	8	8	5	11	6	1	-3	6	5	2	1	1	0	1	1	0	6	7	0	7	3	8	8	5	11	0	3
67	101	67	2	12	61	5	16	16	5	11	1	2	6	15	15	6	9	0	3	6	8	13	13	8	13	8	15	6	5	0	2	4	1	1	2	2	0	0	0	0	0	1	0	3	0	15	6	13	8	2	0
68	101	68	2	4	77	15	1	15	1	14	8	12	7	9	7	9	2	1	2	4	13	3	13	3	7	9	13	3	10	0	6	8	3	2	1	2	1	1	1	2	0	7	1	2	0	7	9	13	3	5	1
69	101	69	2	44	45	1	15	15	1	14	8	12	9	7	7	9	2	1	7	14	8	8	8	8	8	8	7	9	0	1	-1	2	5	1	2	1	1	1	1	2	0	7	1	7	0	7	9	8	8	0	1
70	101	70	2	48	53	15	1	15	1	14	7	14	8	8	8	8	0	1	3	6	12	4	12	4	8	8	12	4	8	0	4	8	4	2	1	2	1	1	1	0	0	7	0	3	0	8	8	12	4	4	0
71	101	71	2	64	19	2	16	16	2	14	11	6	13	5	5	13	8	1	1	2	3	15	15	3	5	13	15	3	12	0	10	4	3	2	2	2	1	1	1	8	0	7	4	1	0	5	13	15	3	6	4
72	101	72	2	16	17	2	15	15	2	13	1	2	3	14	14	3	11	0	10	6	12	5	5	12	5	12	14	3	7	1	9	4	3	1	2	1	1	1	1	0	0	1	0	6.5000	3.5000	14	3	5	12	5.5000	3.5000
73	101	73	2	28	29	1	15	15	1	14	6	12	7	9	9	7	2	0	7	14	8	8	8	8	8	8	9	7	0	1	1	2	4	1	2	1	1	1	1	0	0	6	0	7	0	9	7	8	8	1	0
74	101	74	2	36	37	15	1	15	1	14	7	14	8	8	8	8	0	1	4	8	11	5	11	5	8	8	11	5	6	0	3	6	4	2	1	2	1	1	1	0	0	7	0	4	0	8	8	11	5	3	0
75	101	75	2	18	49	16	2	16	2	14	5	10	11	7	11	7	4	0	6	12	10	8	10	8	10	8	11	7	2	0	1	2	1	1	1	1	1	0	0	0	0	5	0	6	0	11	7	10	8	1	0
76	101	76	2	10	65	2	16	16	2	14	3	6	5	13	13	5	8	0	1	2	3	15	15	3	13	5	15	3	12	0	2	4	1	2	2	1	0	0	0	0	0	3	0	1	0	13	5	15	3	2	0
77	101	77	2	2	3	15	5	15	5	10	4	8	11	9	11	9	2	0	5	10	10	10	10	10	10	10	11	9	0	1	1	2	4	1	1	1	1	1	1	0	0	4	0	5	0	11	9	10	10	1	0
78	101	78	2	32	33	15	1	15	1	14	1	2	14	2	14	2	12	0	2	4	13	3	13	3	13	3	14	2	10	0	1	2	1	1	1	2	0	0	0	0	0	1	0	2	0	14	2	13	3	1	0
79	101	79	2	50	51	3	15	15	3	12	1	2	4	14	14	4	10	0	3	6	6	12	12	6	12	6	14	4	6	0	2	4	1	1	2	2	0	0	0	0	0	1	0	3	0	14	4	12	6	2	0
80	101	80	2	44	49	15	5	15	5	10	6	8	9	11	9	11	2	1	5	10	10	10	10	10	10	10	9	11	0	1	-1	2	5	1	1	2	0	1	1	2	0	5	1	5	0	9	11	10	10	0	1
81	101	81	2	80	79	16	5	16	5	11	1	2	15	6	15	6	9	0	9	4	7	14	7	14	7	14	15	6	7	1	8	2	3	1	1	1	1	1	1	0	7	1	0	5.5000	3.5000	15	6	7	14	4.5000	3.5000
82	101	82	2	22	41	15	3	15	3	12	3	6	12	6	12	6	6	0	8	8	7	11	7	11	7	11	12	6	4	1	5	2	3	1	1	1	1	1	1	0	4	3	0	6	2	12	6	7	11	3	2
83	101	83	2	48	49	3	16	16	3	13	3	6	6	13	13	6	7	0	5	10	8	11	11	8	11	8	13	6	3	0	2	4	1	1	2	2	0	0	0	0	0	3	0	5	0	13	6	11	8	2	0
84	101	84	2	68	67	16	2	16	2	14	9	10	7	11	7	11	4	1	1	2	15	3	15	3	7	11	15	3	12	0	8	8	3	2	1	2	1	1	1	4	0	7	2	1	0	7	11	15	3	6	2
85	101	85	2	32	21	5	16	16	5	11	1	2	6	15	15	6	9	0	8	6	13	8	8	13	8	13	15	6	5	1	7	4	3	1	2	1	1	1	1	0	0	1	0	5.5000	2.5000	15	6	8	13	4.5000	2.5000
86	101	86	2	3	11	15	2	15	2	13	3	6	12	5	12	5	7	0	4	8	11	6	11	6	11	6	12	5	5	0	1	2	1	1	1	1	1	0	0	0	0	3	0	4	0	12	5	11	6	1	0
87	101	87	2	26	31	16	5	16	5	11	1	2	15	6	15	6	9	0	2	4	14	7	14	7	14	7	15	6	7	0	1	2	1	1	1	2	0	0	0	0	0	1	0	2	0	15	6	14	7	1	0
88	101	88	2	26	27	15	3	15	3	12	1	2	14	4	14	4	10	0	2	4	13	5	13	5	13	5	14	4	8	0	1	2	1	1	1	2	0	0	0	0	0	1	0	2	0	14	4	13	5	1	0
89	101	89	2	28	29	16	2	16	2	14	5	10	11	7	11	7	4	0	1	2	15	3	15	3	11	7	15	3	12	0	4	8	1	2	1	1	0	0	0	0	0	5	0	1	0	11	7	15	3	4	0
90	101	90	2	7	31	15	1	15	1	14	6	12	9	7	9	7	2	0	2	4	13	3	13	3	9	7	13	3	10	0	4	8	1	2	1	1	0	0	0	0	0	6	0	2	0	9	7	13	3	4	0
91	101	91	2	64	19	15	1	15	1	14	1	2	14	2	14	2	12	0	12	4	3	13	3	13	3	13	14	2	10	1	11	2	3	1	1	1	1	1	1	0	10	1	0	7	5	14	2	3	13	6	5
92	101	92	2	46	47	16	3	16	3	13	6	12	10	9	10	9	1	0	2	4	14	5	14	5	10	9	14	5	9	0	4	8	1	2	1	2	1	0	0	0	0	6	0	2	0	10	9	14	5	4	0
93	101	93	2	64	65	15	1	15	1	14	7	14	8	8	8	8	0	1	11	6	4	12	4	12	8	8	4	12	8	1	-4	8	5	2	1	1	0	1	1	0	8	7	0	7	4	8	8	4	12	0	4
94	101	94	2	76	77	13	2	13	2	11	6	10	7	8	7	8	1	1	1	2	12	3	12	3	7	8	12	3	9	0	5	8	3	2	1	1	0	1	1	1	0	5.5000	0.5000	1	0	7	8	12	3	4.5000	0.5000
95	101	95	2	24	25	2	16	16	2	14	7	14	9	9	9	9	0	1	5	10	7	11	11	7	9	9	11	7	4	0	2	4	4	2	2	2	1	1	1	0	0	7	0	5	0	9	9	11	7	2	0
96	101	96	2	64	63	3	16	16	3	13	6	12	9	10	10	9	1	0	3	6	6	13	13	6	10	9	13	6	7	0	3	6	1	2	2	1	0	0	0	0	0	6	0	3	0	10	9	13	6	3	0
97	101	97	2	22	21	4	16	16	4	12	3	6	7	13	13	7	6	0	7	10	11	9	9	11	9	11	13	7	2	1	4	4	3	1	2	2	0	1	1	0	0	3	0	6	1	13	7	9	11	3	1
98	101	98	2	20	25	2	16	16	2	14	9	10	11	7	7	11	4	1	2	4	4	14	14	4	7	11	14	4	10	0	7	6	3	2	2	1	0	1	1	4	0	7	2	2	0	7	11	14	4	5	2
99	101	99	2	8	69	3	16	16	3	13	5	10	8	11	11	8	3	0	2	4	5	14	14	5	11	8	14	5	9	0	3	6	1	2	2	1	0	0	0	0	0	5	0	2	0	11	8	14	5	3	0
100	101	100	2	34	35	1	15	15	1	14	4	8	5	11	11	5	6	0	1	2	2	14	14	2	11	5	14	2	12	0	3	6	1	2	2	1	0	0	0	0	0	4	0	1	0	11	5	14	2	3	0
⋮

	SubjectID	a0	b0	a1	b1	a2	b2	Chose1	Prob1
1	101	16	5	14	7	8	13	1	0
2	101	16	1	5	12	15	2	0	0
3	101	16	3	15	4	5	14	0	0
4	101	5	16	11	10	7	14	0	0
5	101	15	2	8	9	13	4	0	0
6	101	5	16	7	14	12	9	1	0
7	101	1	15	12	4	2	14	0	0
8	101	2	15	10	7	4	13	0	0
9	101	15	3	8	10	14	4	0	0
10	101	3	15	10	8	5	13	0	0
11	101	15	1	6	10	14	2	0	0
12	101	3	15	11	7	4	14	1	0
13	101	15	3	11	7	8	10	0	0
14	101	14	2	13	3	4	12	0	0
15	101	15	2	7	10	14	3	0	0
16	101	3	16	12	7	4	15	0	0
17	101	15	2	11	6	7	10	0	0
18	101	2	16	5	13	11	7	0	0
19	101	2	16	4	14	12	6	1	0
20	101	1	16	6	11	9	8	1	0
21	101	3	16	5	14	12	7	1	0
22	101	16	2	13	5	6	12	0	0
23	101	15	3	14	4	5	13	1	0
24	101	3	16	11	8	5	14	0	0
25	101	3	15	4	14	12	6	1	0
26	101	3	16	6	13	11	8	1	0
27	101	16	3	9	10	14	5	0	0
28	101	15	2	12	5	6	11	1	0
29	101	1	15	9	7	4	12	0	0
30	101	15	1	7	9	13	3	1	0
31	101	2	16	13	5	3	15	1	0
32	101	2	15	3	14	12	5	0	0
33	101	16	5	15	6	7	14	0	0
34	101	15	3	12	6	7	11	0	0
35	101	16	2	7	11	15	3	1	0
36	101	5	16	6	15	13	8	0	0
37	101	15	1	14	2	3	13	0	0
38	101	13	2	7	8	12	3	0	0
39	101	4	16	7	13	11	9	1	0
40	101	2	16	11	7	4	14	0	0
41	101	16	2	11	7	8	10	1	0
42	101	1	15	11	5	2	14	0	0
43	101	16	3	8	11	15	4	1	0
44	101	3	16	10	9	6	13	0	0
45	101	2	14	9	7	4	12	0	0
46	101	2	14	10	6	3	13	1	0
47	101	16	2	15	3	4	14	1	0
48	101	3	16	4	15	13	6	1	0
49	101	5	16	12	9	6	15	1	0
50	101	3	15	5	13	11	7	0	0
51	101	16	5	10	11	15	6	1	0
52	101	16	1	7	10	14	3	1	0
53	101	2	15	4	13	11	6	0	0
54	101	1	16	13	4	2	15	0	0
55	101	16	2	8	10	14	4	1	0
56	101	2	16	12	6	3	15	1	0
57	101	15	3	13	5	6	12	0	0
58	101	2	16	6	12	10	8	1	0
59	101	16	2	12	6	7	11	1	0
60	101	16	5	13	8	9	12	1	0
61	101	15	2	14	3	4	13	0	0
62	101	16	1	6	11	15	2	1	0
63	101	14	2	7	9	13	3	0	0
64	101	1	15	3	13	11	5	1	0
65	101	15	2	13	4	5	12	0	0
66	101	2	15	11	6	3	14	1	0
67	102	16	5	14	7	8	13	0	0
68	102	16	1	5	12	15	2	0	0
69	102	16	3	15	4	5	14	1	0
70	102	5	16	11	10	7	14	0	0
71	102	15	2	8	9	13	4	0	0
72	102	5	16	7	14	12	9	0	0
73	102	1	15	12	4	2	14	1	0
74	102	2	15	10	7	4	13	0	0
75	102	15	3	8	10	14	4	1	0
76	102	3	15	10	8	5	13	1	0
77	102	15	1	6	10	14	2	1	0
78	102	3	15	11	7	4	14	1	0
79	102	15	3	11	7	8	10	0	0
80	102	14	2	13	3	4	12	0	0
81	102	15	2	7	10	14	3	1	0
82	102	3	16	12	7	4	15	1	0
83	102	15	2	11	6	7	10	0	0
84	102	2	16	5	13	11	7	0	0
85	102	2	16	4	14	12	6	0	0
86	102	1	16	6	11	9	8	0	0
87	102	3	16	5	14	12	7	1	0
88	102	16	2	13	5	6	12	0	0
89	102	15	3	14	4	5	13	0	0
90	102	3	16	11	8	5	14	1	0
91	102	3	15	4	14	12	6	0	0
92	102	3	16	6	13	11	8	0	0
93	102	16	3	9	10	14	5	1	0
94	102	15	2	12	5	6	11	0	0
95	102	1	15	9	7	4	12	1	0
96	102	15	1	7	9	13	3	1	0
97	102	2	16	13	5	3	15	1	0
98	102	2	15	3	14	12	5	0	0
99	102	16	5	15	6	7	14	0	0
100	102	15	3	12	6	7	11	0	0
⋮

	SubjectID	Deviance	A_F1	A_F2	A_F3	A_F4	A_F5	D_F1	D_F2	D_F3	D_F4	D_F5	B_F1	B_F2	B_F3	B_F4	B_F5	E_F1	E_F2	E_F3	E_F4	E_F5	G_F1	G_F2	G_F3	G_F4	G_F5
1	101	97.1880	0.4782	0.4811	0.4145	1.2103	0.2482	1.4307	1.3962	1.4619	0.7569	1.4869	6.3071	6.0420	5.9012	4.0285	4.2533	0.4764	0.4500	0.4038	0.5000	0.3964	-0.0877	-0.0556	-0.1190	-0.0094	0.0045
2	102	67.7072	1.1095	0.8451	1.5994	0.8706	1.1523	0.1646	0.1202	0.1064	0.1188	0.2034	2.6240	4.0487	7.4646	3.9569	3.3083	0.1347	0.1563	0.1259	0.1010	0.1092	-0.5000	-0.3297	-0.3631	-0.2338	-0.2933
3	103	21.8285	1.5147	1.4924	1.4881	1.4735	1.4882	0.1452	0.1356	0.1368	0.1330	0.1355	6.6214	6.6789	6.6305	6.7560	6.6955	0.0186	4.8216e-05	0.0209	0.0186	0.0200	-0.4981	-0.0162	-0.4984	-0.4981	-0.4984
4	104	21.9339	1.5941	1.5947	1.4771	1.6330	1.5938	0.1011	0.1020	0.1337	0.0745	0.0982	7.4917	7.5030	6.6784	6.3293	7.5542	0.0179	0.0193	4.8248e-05	0.0200	0.0186	0.4980	0.4982	-0.0089	0.4983	0.4981
5	105	22.8919	0.4612	4.0252e-06	3.3869e-06	0.4791	6.5183e-06	1.6334	1.0324	1.0426	1.4377	0.1188	5.1374	0.6397	0.6538	6.5223	5.3681	0.0400	0.0219	0.0298	0.0228	1.3567e-05	-0.4991	-0.5000	-0.5000	-0.4985	-0.3360
6	106	66.9448	1.6152	1.5539	1.6140	1.6016	1.5976	0.1003	0.0908	0.1093	0.0968	0.0987	7.6332	7.5344	7.5096	7.4398	7.6046	0.1896	0.1154	0.1538	0.1923	0.1141	-0.1836	-3.5427e-08	-0.3750	-0.2000	-0.1494
7	107	75.0862	0.2688	0.3756	0.3344	0.2535	0.3125	1.6047	1.6857	1.5068	1.6019	1.5749	3.9619	4.2900	4.1267	4.1073	4.6118	0.2271	0.2607	0.2054	0.1893	0.2464	-0.0283	-0.1164	-0.1957	-0.0283	-0.1243
8	108	93.2399	0.3016	1.9868	0.8220	0.8259	1.1661e-04	0.9049	1.3604	0.6750	0.6462	1.0194	0.0993	9.9382	1.7947	1.7891	0.1750	0.1323	0.3878	0.3550	0.3917	0.4661	-0.4999	-0.1668	-0.2839	-0.1939	-0.1502
9	109	77.7570	1.7492	0.3673	0.2749	0.2087	1.2049e-07	1.7006	1.3090	1.6691	1.5390	0.0594	9.3596	6.1492	3.8712	4.4840	3.7288	0.4422	0.3292	0.3462	0.3264	0.2733	-0.2234	-0.3178	-0.2778	-0.3298	-0.2838
10	110	26.9598	8.7831e-06	3.4212e-06	4.6848e-06	5.3349e-06	0.4314	1.0470	1.0347	1.0426	1.0428	1.5489	0.7270	0.6722	0.6744	0.7269	4.0412	0.0635	0.0538	0.0744	0.0516	0.0417	-0.5000	-0.5000	-0.5000	-0.5000	-0.4992
11	111	0.0069	1.5029	1.4905	1.4874	1.4928	1.4922	0.1277	0.1352	0.1389	0.1399	0.1356	6.1319	6.6638	6.6806	6.6670	6.6815	4.7490e-05	4.7845e-05	4.8289e-05	4.7938e-05	4.7864e-05	0.0954	-0.0182	-0.0139	0.0012	-0.0180
12	112	89.1837	1.1914	0.4065	1.5740	1.6007	0.6154	1.3615	0.8795	1.5888	1.6342	0.7423	7.3283	0.4012	9.6037	9.1971	0.4522	0.2747	0.2322	0.2560	0.2767	0.0976	-0.0141	-0.0816	-0.1663	-0.1917	-0.1630
13	113	0.0068	1.4906	1.4905	1.4903	1.4890	1.4887	0.1395	0.1351	0.1389	0.1362	0.1339	6.6658	6.6642	6.6519	6.6426	6.6381	4.7844e-05	4.7751e-05	4.8447e-05	4.7911e-05	4.7794e-05	-0.0067	-2.9492e-04	-0.0252	-0.0123	-0.0077
14	114	68.9381	1.0594	1.0612	1.0717	1.0767	1.0419	0.1775	0.1086	0.1039	0.1438	3.1824e-06	3.9065	1.5400	1.7983	1.9556	1.0145	0.1569	0.2009	0.1666	0.1865	0.1779	-0.3593	-0.2982	-0.2821	-0.2937	-0.5000
15	115	40.6152	1.4978	1.4820	1.4818	1.4725	1.4789	0.1362	0.1394	0.1337	0.1331	0.1381	6.5665	6.4842	6.6231	6.5591	6.5663	0.1334	0.1667	0.1111	0.1305	0.1400	-0.4997	-0.4998	-0.4997	-0.4998	-0.4998
16	116	31.7668	1.4966	1.4957	1.4899	1.4856	1.5333	0.1365	0.1314	0.1393	0.1370	0.1410	6.6436	6.3717	6.5456	6.5645	6.5403	0.0834	0.0926	0.0800	0.1000	0.0962	-0.4996	-0.4996	-0.4996	-0.4996	-0.4997
17	117	82.4332	0.4378	0.2401	0.2470	1.2504	1.2197	1.5881	1.5238	1.5619	1.9999	1.9859	4.4440	4.2653	4.2184	9.9997	9.9355	0.2711	0.2530	0.1923	0.2792	0.2284	-0.0902	-0.0765	-0.2000	-0.0115	0.0472
18	118	61.2437	1.6063	1.5629	1.6249	1.5749	1.5730	0.0998	0.0984	0.1039	0.0896	0.1018	7.6117	7.4665	7.2526	7.3494	7.3561	0.2084	0.1867	0.2024	0.2381	0.1538	-0.3345	-0.3929	-0.2941	-0.3250	-0.3750
19	119	22.1591	1.5163	1.4921	1.4849	1.4857	1.4923	0.1345	0.1357	0.1518	0.1370	0.1353	6.4528	6.7009	6.6186	6.8713	6.6778	0.0200	4.7720e-05	0.0173	0.0193	0.0209	-0.4984	0.0011	-0.4979	-0.4983	-0.4985
20	120	93.5268	0.5127	1.4516	0.8670	0.5215	1.2887	1.4810	1.9993	1.1172	1.4694	0.8020	4.0034	9.9965	4.1014	3.9969	6.0705	0.5000	0.4799	0.5000	0.5000	0.5000	-0.0192	-0.0849	-0.0769	-0.0962	-0.1538
21	121	88.6196	0.9855	0.6243	1.5058	0.9408	1.0627	0.1609	1.3683	0.0800	1.1176	0.1522	0.3974	3.9935	6.7758	4.1189	1.7240	0.2201	0.5000	0.3155	0.5000	0.3120	-0.4577	-0.1346	-0.1604	-0.2115	-0.3582
22	122	0.0071	1.4753	1.4910	1.4867	1.4870	1.4900	0.1337	0.1358	0.1359	0.1489	0.1352	6.7488	6.6994	6.6392	6.6618	6.6679	4.7765e-05	4.7729e-05	4.8605e-05	4.7861e-05	4.7751e-05	-0.0094	-0.0139	-0.0286	0.0036	-0.0066
23	124	86.4589	2.0000	1.0453	2.0000	0.9810	1.5995	0.5761	0.1638	0.3635	0.1902	0.0945	9.9999	1.6175	9.9999	4.2137	7.3508	0.3056	0.2518	0.2982	0.3013	0.2629	-0.0456	0.0600	-0.0200	0.0517	0.0246
24	125	23.6235	1.2072	1.0957	1.1133	1.1013	1.5930	0.0842	0.0945	0.1056	0.1077	0.1037	3.5874	1.7717	2.5622	2.3558	7.5027	0.0394	4.8474e-06	0.0292	0.0328	0.0186	0.4991	-0.3480	0.4988	0.4992	0.4981
25	126	73.8646	0.9276	1.5760	1.1112	1.5636	1.5743	1.1216	0.0999	0.8560	0.0934	0.0954	4.0569	7.4095	4.1242	7.2248	7.2828	0.5000	0.4167	0.5000	0.3877	0.3681	-0.2255	-0.3000	-0.2885	-0.2937	-0.3491
26	127	91.7783	1.2210	1.6104	1.5936	1.0910	1.6137	0.6805	0.1025	0.0904	0.8544	0.1009	4.0500	7.7714	7.5032	3.9911	7.6468	0.5000	0.3575	0.3654	0.5000	0.4289	-0.0472	-0.0697	-0.1842	-0.1038	-0.0628
27	128	65.5510	1.4908	1.5022	1.4900	1.4915	1.4758	0.1353	0.1379	0.1413	0.1353	0.1350	6.6623	6.7091	6.7314	6.6831	6.5470	0.0536	0.0556	0.0625	4.7776e-05	0.0536	-0.4994	-0.4994	-0.4995	-0.0197	-0.4994
28	129	91.6851	1.2844	1.2599	1.2852	0.2535	1.2613	2.0000	2.0000	2.0000	1.4257	2.0000	9.9999	9.9999	9.9999	4.9598	10.0000	0.3459	0.3304	0.3362	0.2926	0.3889	-0.0545	-0.1048	-0.0743	-0.1835	-0.0707
29	130	21.8617	1.5004	1.4901	1.4918	1.4983	1.5137	0.1397	0.1357	0.1362	0.1406	0.1377	6.7015	6.7038	6.6698	6.6643	6.6641	0.0200	4.8635e-05	0.0179	0.0209	0.0193	-0.4983	-0.0050	-0.4980	-0.4984	-0.4982
30	131	42.6282	1.6121	1.5915	1.5630	1.5916	1.6081	0.1009	0.0993	0.0862	0.1015	0.1012	7.6051	7.5000	7.5072	7.3970	7.5703	0.0957	0.0914	0.0962	0.0914	0.0757	-0.1269	-0.2812	-0.1000	-0.2812	-0.0283
31	132	59.7471	0.3182	1.5791e-05	0.3031	0.4748	0.2581	1.6884	1.0312	1.5223	1.5428	1.4909	4.0137	0.7017	4.8275	6.5969	4.4305	0.1450	0.0816	0.0800	0.1489	0.1339	-0.3621	-0.3464	-0.4996	-0.3708	-0.3617
32	133	85.3685	0.3826	0.5485	0.5299	0.5090	1.4639	0.8860	0.7965	0.8603	0.8237	1.9995	0.3932	0.4938	0.9636	0.3247	9.9977	0.2602	0.2663	0.3355	0.1804	0.3591	-0.2117	-0.4033	-0.2280	-0.5000	-0.2153
33	134	90.5535	1.5747	1.0660	1.0310	0.6950	1.0459	0.0895	0.1845	0.2023	1.2844	0.1373	7.3402	1.8855	1.5330	3.9997	1.3785	0.3252	0.3770	0.3064	0.5000	0.3901	-0.0695	-0.2106	-0.1822	-0.0769	-0.1387
34	135	93.3505	1.4044	1.5199	1.5193	1.2732	1.8612	2.0000	1.9988	1.9977	2.0000	1.9996	10.0000	9.9941	9.9889	9.9998	9.9982	0.3204	0.3361	0.3652	0.3317	0.3571	-0.0920	-0.1878	-0.1346	-0.1985	-0.2000
35	136	55.5598	1.6077	1.5668	1.4696	1.5855	1.5944	0.1012	0.0925	0.1354	0.1041	0.0998	7.4530	7.4181	6.5493	7.5246	7.4839	0.1339	0.1093	0.0962	0.1147	0.0762	-0.3617	-0.3170	-0.4996	-0.3385	-0.2570
36	138	84.0317	0.2412	0.4576	0.2344	0.2616	0.2397	0.6922	1.4281	1.5719	1.4697	1.4939	4.4287	6.6881	4.0390	4.5317	4.1948	0.4403	0.3364	0.3119	0.3452	0.3523	-0.2380	-0.0946	-0.3312	-0.2241	-0.1452
37	140	96.0168	0.4821	0.2943	1.0051	0.9087	1.7132	1.4749	1.5533	1.2505	1.1456	2.0000	6.2952	4.1770	6.2699	5.4545	10.0000	0.4256	0.4048	0.3422	0.3098	0.3201	0.0385	0.0147	0.0179	-0.0233	-0.0371
38	141	91.3465	1.5959	1.5555	1.5170	0.6224	2.0000	1.6082	1.5538	1.5164	1.1730	1.7510	9.0382	8.8107	9.0324	4.1924	10.0000	0.3668	0.4688	0.4005	0.5000	0.4027	-0.1632	-0.0511	-0.1072	-0.0686	-0.0776
39	142	98.5530	1.5810	1.5659	1.5344	0.6930	1.9987	0.0933	0.0943	0.0893	1.2839	1.4004	7.5324	7.3812	7.2323	3.9867	9.9934	0.3967	0.4145	0.4137	0.5000	0.4116	-0.0332	-0.0361	-0.1475	-0.0283	-0.2004
40	143	86.4086	0.4776	1.5969	1.5396	1.0148	1.6214	1.5136	0.0935	0.0887	0.9540	0.0906	3.9862	7.4591	7.5123	4.1221	7.4984	0.5000	0.2692	0.2467	0.5000	0.2197	-0.1154	-0.2857	-0.1757	-0.1346	-0.1207
41	144	21.5113	1.4838	1.4858	1.4974	1.4928	1.4921	0.1372	0.1393	0.1344	0.1341	0.1359	6.6345	6.6116	6.6086	6.6158	6.6494	0.0193	0.0179	0.0209	0.0209	4.9496e-05	-0.4982	-0.4980	-0.4984	-0.4984	-0.0319
42	145	89.7992	0.6279	1.5407	2.0000	1.9997	1.8272	1.2021	1.5086	0.6097	1.8511	1.7656	5.3102	8.5852	9.9998	9.9987	9.5203	0.5000	0.3918	0.3365	0.4072	0.4254	-0.1400	-0.1666	-0.2654	-0.2368	-0.2103
43	146	93.6328	0.5742	0.3459	2.0000	2.0000	2.0000	1.4195	1.0964	1.5484	1.6518	1.6486	4.0128	2.9493	9.9999	9.9999	9.9999	0.5000	0.4793	0.4007	0.4078	0.4269	-0.1415	-0.2079	-0.1872	-0.2242	-0.2028
44	147	0.0073	0.4515	0.4663	0.5652	0.4783	0.6312	1.3664	1.3579	1.3356	1.3447	1.4665	4.1947	4.1123	5.8132	4.1445	5.3348	4.8934e-05	5.6209e-05	5.4811e-05	4.8731e-05	5.4675e-05	-0.0055	0.0365	-0.0047	9.4005e-04	0.0123
45	148	34.1181	1.4920	1.5059	1.4829	1.5587	1.4745	0.1382	0.1337	0.1343	0.1493	0.1301	6.7043	6.5376	6.5376	6.6335	6.5564	0.0834	0.0600	0.1154	0.0968	0.0962	-0.4996	-0.4995	-0.4997	-0.4996	-0.4997
46	149	22.6940	2.0000	1.9997	1.9997	1.9995	1.4898	0.3812	0.3635	0.3635	0.3635	0.1388	9.9999	9.9986	9.9986	9.9973	6.6615	3.2270e-06	3.0099e-05	2.9809e-05	4.2908e-05	4.7838e-05	-0.0168	-0.3240	-0.3212	-0.2759	0.0060
47	150	80.6691	0.5798	0.5230	0.1223	0.3117	1.4046	0.9900	0.8480	0.9945	1.4075	2.0000	2.8508	3.9132	0.1522	4.0551	9.9999	0.2989	0.2881	0.0731	0.3359	0.3198	-0.1814	-0.2926	-0.5000	-0.1617	-0.2010
48	151	63.0235	1.5817	1.5693	1.6390	1.5880	1.5885	0.0851	0.0947	0.0903	0.0932	0.0987	7.3271	7.5587	7.3767	7.5200	7.5251	0.2197	0.2319	0.2352	0.2082	0.2121	-0.1207	-0.2187	-0.0951	-0.1725	-0.2857
49	152	67.9910	1.6319	1.0251	1.0526	1.0916	1.0577	0.0907	0.0139	0.0738	1.6190e-06	4.8798e-06	7.4082	0.8338	1.1882	1.0874	1.0527	0.1887	0.1451	0.1435	0.2115	0.1896	-0.0094	-0.1382	-0.1618	-0.1102	-0.0828
50	153	101.9017	0.2999	0.3475	0.3507	0.3730	1.8361	1.5610	1.6243	1.4413	1.5928	1.9985	4.4382	4.0062	4.1418	4.6515	9.9924	0.3768	0.4131	0.4422	0.3521	0.4870	-0.0406	0.1276	0.0025	0.1592	0.0436
51	154	42.2913	1.5975	1.5054	1.6039	1.5978	1.6022	0.0953	0.1351	0.1061	0.1021	0.1000	7.3475	6.6170	7.5307	7.5005	7.5803	0.0377	0.0186	0.0387	0.0179	0.0381	0.0094	-0.4982	-0.0385	0.4980	0.0472
52	155	49.1139	2.0000	2.0000	2.0000	1.5911	2.0000	0.3788	0.3627	0.3624	0.1023	0.3618	10.0000	10.0000	10.0000	7.4850	9.9998	0.0337	0.0573	0.0603	0.0377	0.0414	0.0504	0.1373	0.1820	-0.0094	0.4994
53	156	69.6066	0.4997	0.4980	0.4844	0.4700	0.4542	0.9635	0.9625	0.9479	1.4434	0.9238	2.2276	2.2272	1.8983	6.5570	1.7022	0.0411	0.0382	0.0361	0.0371	0.0019	-0.3919	-0.3501	-0.4648	-0.4991	0.4999
54	157	116.6987	1.0125	2.0000	1.9910	1.9983	2.0000	1.9405e-06	0.7502	1.3476	1.3737	0.7508	0.2868	10.0000	9.9602	9.9916	10.0000	0.1600	0.2869	0.2647	0.2300	0.3318	-0.5000	-0.1365	-0.3158	-0.5000	-0.1669
55	158	92.9659	0.6404	1.2778	0.2527	1.2454	1.2904	1.2054	2.0000	1.5301	0.7841	2.0000	4.0910	9.9998	4.1191	3.9834	9.9999	0.5000	0.4462	0.4375	0.5000	0.4643	-0.0660	-0.0789	-0.0714	-0.0660	-0.0572
56	160	59.1946	0.4923	0.3168	0.4957	0.2625	0.3483	1.4553	1.5504	1.4499	1.3884	1.5504	6.7061	4.7091	6.3644	5.0101	4.8179	0.1556	0.1731	0.1711	0.1156	0.1526	0.1429	0.0556	0.1494	-0.0192	0.1068
57	161	90.5272	0.4890	1.3000	1.4186	0.3344	0.2593	1.4212	0.4209	0.4246	1.6068	1.5453	6.4144	4.0970	4.0726	4.5608	4.2288	0.3846	0.5000	0.5000	0.3720	0.3958	-0.2000	-0.1923	-0.2115	-0.2200	-0.1316

	SubjectID	Alpha_M1	Beta_M1	Epsilon_M1	Gamma_M1	SS_M1	Deviance_M1	Delta_M2	Beta_M2	Epsilon_M2	Gamma_M2	SS_M2	Deviance_M2	Rho_M3	Beta_M3	Epsilon_M3	Gamma_M3	SS_M3	Deviance_M3	Alpha_M4	Delta_M4	Beta_M4	Epsilon_M4	Gamma_M4	SS_M4	Deviance_M4	Alpha_M5	Rho_M5	Beta_M5	Epsilon_M5	Gamma_M5	SS_M5	Deviance_M5	Delta_M6	Rho_M6	Beta_M6	Epsilon_M6	Gamma_M6	SS_M6	Deviance_M6	Alpha_M7	Delta_M7	Rho_M7	Beta_M7	SS_M7	Deviance_M7	Alpha_M8	Delta_M8	Rho_M8	Beta_M8	Epsilon_M8	SS_M8	Deviance_M8	Gamma_M9	SS_M9	Deviance_M9
1	101	0.9996	4.0204	0.5000	-0.0455	16.3636	90.9492	1.1755	4.0766	0.4394	-0.0517	16.1212	89.9689	0.9239	3.9647	0.4362	-0.0521	16.1212	89.9689	0.2345	1.4538	4.3429	0.4394	-0.0517	16.1212	89.9689	0.9876	1.0059	4.0166	0.5000	-0.0455	16.3636	90.9492	1.0000	1.0000	4.0184	0.4394	-0.0517	16.1212	89.9689	5.2475e-07	2.4427e-07	1.3479	0.1807	16.2576	90.5233	0.4412	1.4597	1.0691	4.1116	0.4394	16.2576	90.5233	-0.0455	16.5000	91.4954
2	102	1.6295	4.3651	0.1364	-0.2778	7.3939	49.1954	1.0000	4.0381	0.5000	-0.0758	16.1212	89.9744	0.1916	3.5848	0.5000	-0.0758	16.1212	89.9744	1.1513	0.1651	3.1218	0.1256	-0.3444	7.3523	48.5729	1.4784	0.4183	5.1971	0.1364	-0.2778	7.3939	49.1954	1.0000	1.0000	4.0381	0.5000	-0.0758	16.1212	89.9744	1.3839	0.1870	4.5093e-04	2.1135	8.8712	284.2021	1.6039	0.0858	0.3961	7.6261	0.1364	7.7727	52.5765	-0.0758	16.5000	91.4954
3	103	1.3698	4.1702	0.0157	-0.4979	0.9688	8.9047	0.9999	4.0269	0.5000	-0.0231	16.2154	89.9706	0.1024	3.4158	0.5000	-0.0231	16.2154	89.9706	1.4862	0.1328	6.6532	0.0157	-0.4981	0.9688	8.9041	1.3381	0.8134	6.3171	0.0157	-0.4978	0.9688	8.9046	1.0000	1.0000	4.0269	0.5000	-0.0231	16.2154	89.9706	1.5480	0.0728	0.4774	5.0098	1.0000	46.0531	1.6714	0.0792	0.5377	7.7496	0.0154	0.9846	10.3333	-0.0231	16.2500	90.1091
4	104	0.5114	2.7783	3.2654e-06	-0.1373	0.9447	7.8535	0.9999	4.0163	0.5000	0.0152	16.4849	91.4348	0.0237	1.4694	0.4974	0.0152	16.4877	91.4464	1.5914	0.0989	7.5094	0.0152	0.4980	0.9697	8.9665	0.8776	1.0586	4.0682	5.0665e-05	0.0521	0.9444	7.7305	1.0000	1.0000	4.0163	0.5000	0.0152	16.4849	91.4348	1.1663	0.0063	0.9174	2.0908	0.9486	7.8216	1.6418	0.0691	0.5621	7.5030	0.0152	0.9848	10.3641	0.0152	16.5000	91.4954
5	105	0.9905	4.0225	0.5000	-0.0312	15.9375	88.4727	0.1820	4.0823	0.0208	-0.5000	1.8006	14.8022	1.5068	7.5555	0.0313	-0.4991	1.8750	14.9671	4.0444e-06	1.0325	0.7197	0.0208	-0.5000	1.8006	14.8022	0.0278	1.8204	9.0139	0.0313	-0.4990	1.8750	14.9667	0.2037	3.2526e-06	3.6486	0.0208	-0.5000	1.8006	14.8022	1.8653e-06	0.7461	1.0950e-05	0.8598	1.8414	18.8393	0.4795	1.3870	1.1075	4.4181	0.0313	1.9375	17.7998	-0.0312	16	88.7228
6	106	1.3212	6.6075	0.1529	-0.1935	8.2339	54.0180	1.0000	4.0260	0.5000	-0.0538	16.0615	89.3538	8.4391e-04	3.3499	0.4994	-0.0539	16.0616	89.3540	1.6109	0.0986	7.5899	0.1529	-0.1935	8.2339	54.0180	1.6153	0.6099	8.0110	0.1529	-0.1935	8.2339	54.0180	1.0000	1.0000	4.0260	0.5000	-0.0538	16.0615	89.3538	1.6502	0.0855	0.5905	4.7771	10.0000	473.7453	1.7022	0.0740	0.5134	8.0040	0.1538	8.4615	55.8120	-0.0538	16.2500	90.1091
7	107	0.9928	4.0249	0.5000	-0.0455	16.3636	90.9492	1.3787	4.2427	0.2273	-0.1000	11.4545	69.9661	1.0175	4.0378	0.2182	-0.1042	11.4545	69.9661	0.2827	1.6190	4.0534	0.2273	-0.1000	11.4545	69.9661	0.0171	1.7208	8.2653	0.2273	-0.1000	11.4545	69.9661	1.1538	1.0513	4.1527	0.2273	-0.1000	11.4545	69.9661	5.0894e-08	3.1019e-07	0.3182	3.8459	11.5909	70.7467	0.3709	1.3523	1.2430	4.2690	0.2273	11.5909	70.7467	-0.0455	16.5000	91.4954
8	108	1.0011	4.0623	0.5000	-0.1061	15.7576	88.5031	0.0117	3.6371	0.1054	-0.4997	15.5243	87.5555	0.9529	4.7601	0.4690	-0.1131	15.6970	88.2490	0.7953	0.6573	1.3950	0.3935	-0.2040	15.4957	87.4619	1.0139	0.9928	4.0136	0.5000	-0.1061	15.7576	88.5031	1.0000	1.0000	4.0247	0.4697	-0.1129	15.6970	88.2490	2.0441e-06	1.3345	1.0884e-05	0.0257	16.2573	90.5192	1.4188	0.2956	0.8211	4.1554	0.5000	16.5000	91.4954	-0.1061	16.5000	91.4954
9	109	0.9849	4.0662	0.5000	-0.1875	13.7500	79.4991	1.1672	4.0760	0.3437	-0.2727	12.1875	71.9739	0.9957	4.9879	0.3416	-0.2745	12.1875	71.9739	0.3395	1.6147	4.4653	0.3437	-0.2727	12.1875	71.9739	0.2387	1.9999	9.9996	0.3209	-0.3277	12.1053	71.3591	1.0000	1.0000	4.0047	0.3437	-0.2727	12.1875	71.9739	6.3150e-09	0.0095	0.0575	5.2776	14.4047	82.2280	0.2892	1.3199	1.0511	4.1861	0.3437	14.4375	82.3669	-0.1875	16	88.7228
10	110	0.9897	4.0158	0.5000	-0.0645	15.2419	84.9151	0.2642	3.0256	0.0577	-0.5000	3.4388	24.0005	1.4975	7.4873	0.0645	-0.4995	3.4839	23.8460	5.4358e-06	1.0350	0.7723	0.0577	-0.5000	3.4388	24.0005	0.0294	1.8167	8.9877	0.0645	-0.4995	3.4839	23.8457	1.0189	1.3708e-05	0.7845	0.0577	-0.5000	3.4388	24.0005	4.1573e-07	0.5517	3.1257e-05	0.9153	3.7325	32.3045	0.5595	1.3154	1.0718	5.8812	0.0645	3.7419	29.6629	-0.0645	15.5000	85.9503
11	111	1.1149	5.5743	5.4500e-05	-0.0034	2.0413e-07	0.0073	0.9999	3.9821	0.5000	1.4217e-08	16.5000	91.4954	0.0343	3.4461	0.4991	-7.3164e-06	16.5037	91.5102	1.4907	0.1352	6.6739	4.2872e-05	-0.0111	1.3984e-07	0.0060	1.3317	0.8195	6.2815	4.7994e-05	-0.0087	1.6948e-07	0.0067	1.0000	1.0000	3.9900	0.5000	3.4442e-09	16.5000	91.4954	1.5810	0.0989	0.6268	6.4028	2.4791e-08	8.0190e-04	1.5837	0.1030	0.6234	6.9733	4.2490e-05	1.4420e-07	0.0060	0	16.5000	91.4954
12	112	0.9910	4.0148	0.5000	0.0231	16.2154	89.9706	0.0897	1.2421	0.0269	0.4998	14.1811	81.5138	0.9407	3.9962	0.3303	0.0387	14.5114	83.0113	1.6265	1.6633	9.4870	0.2840	-0.1296	13.9254	80.4270	0.2530	1.9999	9.9996	0.2865	-0.0453	14.0606	81.0658	0.2032	1.3894e-07	0.5482	0.0269	0.5000	14.1811	81.5138	0.7154	1.1587	0.1662	0.1540	13.9814	80.7336	1.2611	1.3343	4.5506e-06	6.4642	0.3197	14.1964	81.7087	0.0231	16.2500	90.1091
13	113	1.1149	5.5743	5.4500e-05	-0.0034	2.0413e-07	0.0073	0.9999	3.9821	0.5000	1.4217e-08	16.5000	91.4954	0.0343	3.4461	0.4991	-7.3164e-06	16.5037	91.5102	1.4907	0.1352	6.6739	4.2872e-05	-0.0111	1.3984e-07	0.0060	1.3317	0.8195	6.2815	4.7994e-05	-0.0087	1.6948e-07	0.0067	1.0000	1.0000	3.9900	0.5000	3.4442e-09	16.5000	91.4954	1.5810	0.0989	0.6268	6.4028	2.4791e-08	8.0190e-04	1.5837	0.1030	0.6234	6.9733	4.2490e-05	1.4420e-07	0.0060	0	16.5000	91.4954
14	114	0.3343	3.8320	0.1835	-0.3244	9.6812	60.3935	1.0000	4.0875	0.5000	-0.1061	15.7576	88.5031	1.8101e-07	2.6161	0.2888	-0.1836	15.7576	88.5031	1.0794	0.1215	1.9524	0.1824	-0.3305	9.6603	60.1663	0.9123	0.5666	2.0255	0.1841	-0.3225	9.6807	60.3904	1.0000	1.0000	4.0875	0.5000	-0.1061	15.7576	88.5031	1.3377	5.1408e-07	3.8893e-06	0.2297	12.4504	79.7345	1.7054	0.0704	0.5766	7.6246	0.1970	10.4394	65.4948	-0.1061	16.5000	91.4954
15	115	1.1171	5.5851	0.1364	-0.4997	6.5455	38.6775	1.0000	4.1159	0.5000	-0.1364	15.2727	86.5236	3.6313e-07	5.4907	0.4574	-0.1491	15.2727	86.5236	1.4749	0.1344	6.6421	0.1364	-0.4998	6.5455	38.6771	1.3010	0.6052	5.6658	0.1364	-0.4998	6.5455	38.6776	1.0000	1.0000	4.1160	0.5000	-0.1364	15.2727	86.5236	1.5882	0.0986	0.6450	6.5117	9.0000	414.4660	1.6860	0.0700	0.5464	8.0086	0.1364	7.7727	52.5765	-0.1364	16.5000	91.4954
16	116	1.1226	5.6129	0.0909	-0.4996	4.9091	31.2978	1.0000	4.0818	0.5000	-0.0909	15.9545	89.3014	5.8531e-04	3.0748	0.4961	-0.0916	15.9547	89.3022	1.4751	0.1327	6.6471	0.0909	-0.4997	4.9091	31.2974	1.2990	0.6642	6.0169	0.0909	-0.4997	4.9091	31.2974	1.0000	1.0000	4.0819	0.5000	-0.0909	15.9545	89.3014	1.5460	0.0611	0.3607	4.4132	6.0000	276.3120	1.7056	0.0762	0.5378	7.9671	0.0909	5.4545	40.2120	-0.0909	16.5000	91.4954
17	117	0.9932	4.0152	0.5000	-0.0231	16.2154	89.9706	1.3671	4.2384	0.2457	-0.0549	12.0142	72.2934	0.9825	4.0150	0.2357	-0.0573	12.0142	72.2934	0.2464	1.3479	5.0595	0.2457	-0.0549	12.0142	72.2934	0.2528	2.0000	10.0000	0.1623	-0.3398	10.9276	65.6919	1.0933	1.0334	4.1026	0.2457	-0.0549	12.0142	72.2934	4.5155e-07	4.2333e-07	1.3526	0.8274	12.0615	72.5491	0.4163	1.4602	1.0436	4.1510	0.2462	12.0615	72.5491	-0.0231	16.2500	90.1091
18	118	1.6875	4.2575	0.1979	-0.3421	9.2083	56.9726	1.0000	4.0972	0.5000	-0.1308	15.1385	85.6109	1.1343e-07	3.1783	0.2174	-0.3008	15.1385	85.6109	1.5912	0.0937	7.4290	0.1979	-0.3421	9.2083	56.9726	1.4923	0.7512	6.9552	0.1979	-0.3421	9.2083	56.9726	1.0000	1.0000	4.0972	0.5000	-0.1308	15.1385	85.6109	1.3512	4.1967e-07	1.7084e-05	0.9702	12.9343	233.4483	1.6951	0.0689	0.4863	7.9353	0.2000	10.4000	65.0523	-0.1308	16.2500	90.1091
19	119	1.3472	4.1692	0.0152	-0.4977	0.9697	8.9674	0.9999	4.0163	0.5000	-0.0152	16.4849	91.4348	0.0230	1.4873	0.4974	-0.0152	16.4877	91.4464	1.4934	0.1351	6.6885	0.0152	-0.4980	0.9697	8.9667	1.6010	0.6762	4.2036	0.0152	-0.4979	0.9697	8.9674	1.0000	1.0000	4.0163	0.5000	-0.0152	16.4849	91.4348	1.5666	0.0883	0.4864	5.5850	1.0000	46.0526	1.6605	0.0813	0.5087	7.7868	0.0152	0.9848	10.3641	-0.0152	16.5000	91.4954
20	120	1.7779	4.2365	0.3987	-0.1081	15.1174	85.4669	1.0000	4.0899	0.5000	-0.0846	15.7846	88.2386	0.0417	3.5751	0.4998	-0.0847	15.7848	88.2394	1.2502	0.7677	3.8669	0.5000	-0.0846	15.7846	88.2386	1.7563	0.2568	4.8530	0.3987	-0.1081	15.1174	85.4669	1.0000	1.0000	4.0899	0.5000	-0.0846	15.7846	88.2386	1.0434	0.5297	0.5534	3.3489e-08	16.2500	90.1091	1.2389	0.6792	0.9815	3.7933	0.5000	16.2500	90.1091	-0.0846	16.2500	90.1091
21	121	1.5760	4.2096	0.3409	-0.2333	12.9091	75.6546	1.0000	4.0593	0.5000	-0.1615	14.5538	83.2013	5.2627e-08	3.5598	0.2926	-0.2760	14.5538	83.2013	1.1221	0.1133	2.3905	0.3376	-0.2405	12.9073	75.6417	1.5335	0.8103	6.9186	0.3409	-0.2333	12.9091	75.6546	1.0000	1.0000	4.0594	0.5000	-0.1615	14.5538	83.2013	0.0278	8.1461e-09	6.3605e-08	3.1028	15.4781	87.0211	0.3418	1.6116	1.0474	4.0246	0.5000	16.2500	90.1091	-0.1615	16.2500	90.1091
22	122	1.1149	5.5742	5.4500e-05	-0.0034	2.0413e-07	0.0073	0.9999	3.9821	0.5000	1.3140e-08	16.5000	91.4954	0.0342	3.4454	0.4990	-7.7564e-06	16.5037	91.5102	1.4906	0.1352	6.6738	4.2872e-05	-0.0111	1.3984e-07	0.0060	1.3320	0.8196	6.2799	4.7989e-05	-0.0084	1.6945e-07	0.0067	1.0000	1.0000	3.9891	0.5000	9.3726e-09	16.5000	91.4955	1.5810	0.0989	0.6268	6.4028	2.4791e-08	8.0190e-04	1.5847	0.1025	0.6306	6.9747	4.2481e-05	1.4425e-07	0.0060	0	16.5000	91.4954
23	124	1.2497	4.0880	0.3030	0.0500	13.8788	80.6828	1.0000	4.0340	0.5000	0.0303	16.4394	91.2529	9.8594e-04	3.4509	0.4991	0.0304	16.4394	91.2530	1.0722	0.1672	2.0452	0.2855	0.0222	13.8194	80.4151	1.4059	0.7961	4.0925	0.3030	0.0500	13.8788	80.6828	1.0000	1.0000	4.0341	0.5000	0.0303	16.4394	91.2529	1.3689	5.2492e-06	4.0752e-06	0.1207	13.9532	80.9403	1.3481	0.2199	2.5344e-04	1.4982	0.2815	13.8267	80.4517	0.0303	16.5000	91.4954
24	125	1.6626	4.4897	0.0303	0.4990	1.8788	15.0938	0.9999	4.0267	0.5000	0.0303	16.4394	91.2529	0.1019	3.4158	0.5000	0.0303	16.4394	91.2529	1.1292	0.1028	2.7437	0.0256	0.4988	1.8740	15.0043	1.4756	0.7350	7.0459	0.0303	0.4989	1.8788	15.0943	1.0000	1.0000	4.0267	0.5000	0.0303	16.4394	91.2529	0.6230	0.0512	0.1259	3.7366	1.9210	21.4511	1.3595	0.1236	1.5615e-04	1.6898	0.0178	1.9047	17.0056	0.0303	16.5000	91.4954
25	126	0.3272	3.8701	0.3935	-0.3193	11.3352	68.3571	1.0000	4.0769	0.5000	-0.2500	12.0000	71.9789	0.0990	2.8793	0.4680	-0.2673	12.0601	72.2997	1.1186	0.8748	4.1186	0.5000	-0.2500	12.0000	71.9789	0.8690	1.5185	7.8874	0.3892	-0.3287	11.3301	68.3247	1.0000	1.0000	4.0769	0.5000	-0.2500	12.0000	71.9789	1.3511	1.7080e-06	1.4988e-05	0.0068	15.9909	88.6865	0.4929	1.2089	0.8543	3.9881	0.5000	16.0000	88.7228	-0.2500	16	88.7228
26	127	1.3199	6.5976	0.3788	-0.1000	15.1515	85.9597	1.0000	4.1108	0.5000	-0.0758	16.1212	89.9744	2.1886e-04	3.5074	0.4997	-0.0758	16.1212	89.9744	1.6011	0.0922	7.5940	0.3788	-0.1000	15.1515	85.9597	1.6127	0.4302	7.2717	0.3788	-0.1000	15.1515	85.9597	1.0000	1.0000	4.1109	0.5000	-0.0758	16.1212	89.9744	1.3410	4.7227e-07	4.3497e-06	0.0199	16.4206	91.1807	0.7917	1.0537	1.0063	4.0090	0.5000	16.5000	91.4954	-0.0758	16.5000	91.4954
27	128	1.1105	5.4754	0.0455	-0.4992	2.7273	20.1109	0.9999	4.0259	0.5000	-0.0455	16.3636	90.9492	0.0059	3.2632	0.4997	-0.0455	16.3638	90.9500	1.5374	0.1434	6.6391	0.0455	-0.4993	2.7273	20.1102	1.3731	0.5326	5.0621	0.0455	-0.4993	2.7273	20.1108	1.0000	1.0000	4.0259	0.5000	-0.0455	16.3636	90.9492	1.6076	0.0710	0.5459	4.9157	3.0000	138.1563	1.6839	0.0764	0.5002	7.8819	0.0455	2.8636	24.4078	-0.0455	16.5000	91.4954
28	129	1.0015	4.0834	0.5000	-0.0538	16.0615	89.3538	1.2775	4.1303	0.3546	-0.0728	14.6884	83.7148	0.9474	4.0054	0.3480	-0.0742	14.6884	83.7148	1.8421	2.0000	10.0000	0.3068	-0.2282	14.1189	80.8510	0.9305	1.0349	4.0088	0.5000	-0.0538	16.0615	89.3538	1.0000	1.0000	4.0383	0.3546	-0.0728	14.6884	83.7148	0.1214	0.3734	1.1635	0.1777	15.0654	85.3551	0.4476	0.2130	2.0000	9.9998	0.3165	14.2687	82.0730	-0.0538	16.2500	90.1091
29	130	1.3685	4.1780	0.0157	-0.4978	0.9688	8.9046	0.9999	4.0163	0.5000	-0.0156	15.9844	88.6603	0.0469	3.3388	0.5000	-0.0156	15.9846	88.6611	1.4806	0.1398	6.9681	0.0157	-0.4981	0.9688	8.9041	1.6125	0.6701	4.2098	0.0157	-0.4979	0.9688	8.9048	1.0000	1.0000	4.0163	0.5000	-0.0156	15.9844	88.6603	1.5779	0.1042	0.6468	6.8897	1.0000	46.0524	1.6636	0.0772	0.5247	7.6633	0.0156	0.9844	10.3021	-0.0156	16	88.7228
30	131	1.7653	4.6747	0.0909	-0.1667	5.3939	39.4657	0.9999	4.0276	0.5000	-0.0303	16.4394	91.2529	0.0168	2.8024	0.4973	-0.0307	16.4428	91.2664	1.5988	0.0942	7.5168	0.0909	-0.1667	5.3939	39.4657	1.5293	0.6593	7.4129	0.0909	-0.1667	5.3939	39.4657	1.0000	1.0000	4.0276	0.5000	-0.0303	16.4394	91.2529	1.6006	0.0863	0.6660	5.9401	6.0000	285.5214	1.6630	0.0742	0.4817	8.0497	0.0909	5.4545	40.2120	-0.0303	16.5000	91.4954
31	132	0.9965	4.0807	0.5000	-0.0909	15.9545	89.3014	0.3469	3.9380	0.1201	-0.3758	6.4822	43.0653	1.1649	5.5148	0.1200	-0.3788	6.4848	43.0681	0.4803	1.4681	6.7319	0.1212	-0.3750	6.4848	43.0681	0.2211	1.9994	9.9970	0.1061	-0.5000	6.4438	41.3417	1.0000	1.0000	4.0995	0.1212	-0.3750	6.4848	43.0681	2.5178e-08	1.1479e-07	0.4331	4.5742	7.0303	48.7520	0.5791	1.4915	0.9804	5.3504	0.1212	7.0303	48.7520	-0.0909	16.5000	91.4954
32	133	0.9972	4.0112	0.5000	-0.1364	15.2727	86.5236	0.0476	2.4669	0.1340	-0.5000	13.9076	80.5246	0.8999	3.9821	0.3719	-0.1834	14.3030	82.2720	0.4854	0.8246	0.3397	0.2110	-0.3937	13.8225	80.5423	0.0158	1.7445	8.4715	0.3788	-0.1800	14.3030	82.2721	0.0867	3.4871e-08	1.3538	0.1340	-0.5000	13.9076	80.5246	7.4038e-07	1.3386	6.4444e-06	0.0634	15.0985	85.7632	4.6123e-06	1.3342	4.0143e-05	0.0637	8.3208e-05	15.0985	85.7632	-0.1364	16.5000	91.4954
33	134	0.2784	3.7897	0.3565	-0.1346	14.6201	83.3339	1.0000	3.9757	0.5000	-0.0846	15.7846	88.2386	0.0013	3.4811	0.4952	-0.0855	15.7848	88.2394	0.3047	0.0215	4.1617	0.3549	-0.1377	14.6177	83.3200	0.8693	1.6731	8.8258	0.3344	-0.1796	14.5390	82.9374	1.0000	1.0000	3.9757	0.5000	-0.0846	15.7846	88.2386	1.3559	4.9393e-07	4.6904e-06	0.0343	16.0232	89.2267	0.7350	1.1587	0.9122	4.0384	0.5000	16.2500	90.1091	-0.0846	16.2500	90.1091
34	135	1.0019	4.0812	0.5000	-0.0692	15.9385	88.8590	1.2789	4.1199	0.3835	-0.0926	15.0568	85.2241	0.9684	4.0389	0.3788	-0.0938	15.0568	85.2241	1.5017	1.9960	9.9867	0.3549	-0.1499	14.7627	83.8164	0.8694	1.0663	3.9969	0.5000	-0.0692	15.9385	88.8590	1.1046	1.0352	4.0900	0.3835	-0.0926	15.0568	85.2241	2.0210e-07	3.5272e-07	1.2842	0.3660	15.3846	86.6162	0.3310	0.1087	1.9998	9.9990	0.3629	15.2218	85.9427	-0.0692	16.2500	90.1091
35	136	1.3006	6.6521	0.1061	-0.3571	5.8788	40.2556	1.0000	4.0377	0.5000	-0.0758	16.1212	89.9744	0.1901	3.4870	0.5000	-0.0758	16.1212	89.9744	1.5823	0.0975	7.4361	0.1061	-0.3571	5.8788	40.2556	1.5240	0.3111	6.0326	0.1061	-0.3571	5.8788	40.2556	1.0000	1.0000	4.0377	0.5000	-0.0758	16.1212	89.9744	1.5307	0.0804	0.3959	5.1966	7.0000	326.9682	1.7330	0.0577	0.6606	7.6153	0.1061	6.2576	44.6423	-0.0758	16.5000	91.4954
36	138	0.9988	4.0106	0.5000	-0.1379	13.3966	75.9335	1.3280	4.1072	0.3571	-0.2000	12.2143	70.6853	0.9278	3.9878	0.3499	-0.2041	12.2143	70.6853	0.3335	1.5839	4.5184	0.3571	-0.2000	12.2143	70.6853	0.9086	1.0451	4.0016	0.5000	-0.1379	13.3966	75.9335	1.0000	1.0000	4.0213	0.3571	-0.2000	12.2143	70.6853	2.6759e-09	4.9825e-09	1.3486	0.4200	13.3966	75.9335	0.4939	1.5546	1.1010	4.0685	0.3621	13.3966	75.9335	-0.1379	14.5000	80.4051
37	140	0.9985	4.0233	0.5000	0.0385	16.1538	89.7241	1.3793	4.1535	0.4006	0.0461	15.5114	87.1219	0.8853	3.9692	0.3945	0.0468	15.5114	87.1219	0.8087	1.3110	6.2211	0.3952	0.0400	15.4912	87.0408	0.6520	1.1702	3.9787	0.5000	0.0385	16.1538	89.7241	1.2526	1.0595	4.1428	0.4006	0.0461	15.5114	87.1219	0.0677	3.0698e-07	1.3422	0.3983	15.5871	87.4391	1.5033	1.5452	1.9997	9.9987	0.3692	15.1388	85.6121	0.0385	16.2500	90.1091
38	141	1.3041	6.5154	0.4768	-0.0412	15.6169	86.8030	1.0000	4.0192	0.5000	-0.0397	15.6508	86.9393	0.0124	3.5995	0.4888	-0.0406	15.6515	86.9422	1.8446	1.9998	9.9991	0.4265	-0.0862	15.3828	85.8458	1.4720	0.5949	6.9554	0.4768	-0.0412	15.6169	86.8030	1.0000	1.0000	4.0192	0.5000	-0.0397	15.6508	86.9393	0.8182	0.2485	0.8958	5.0677e-07	15.7500	87.3366	1.6831	0.0729	0.5111	7.4509	0.4762	15.7143	87.1936	-0.0397	15.7500	87.3365
39	142	1.3028	6.5149	0.4242	-0.0714	15.8788	88.9792	1.0000	4.0514	0.5000	-0.0606	16.2576	90.5233	0.0448	1.7438	0.4684	-0.0648	16.2700	90.5739	1.5743	0.0900	7.2495	0.4242	-0.0714	15.8788	88.9792	1.4040	0.5811	7.4204	0.4242	-0.0714	15.8788	88.9792	1.0000	1	4.0514	0.5000	-0.0606	16.2576	90.5233	1.3305	1.0211e-06	1.1015e-05	0.0199	16.4211	91.1806	0.6911	1.0701	1.0675	4.0015	0.5000	16.5000	91.4954	-0.0606	16.5000	91.4954
40	143	1.3048	6.5531	0.2633	-0.2122	11.7424	70.4117	1.0000	4.0810	0.5000	-0.1154	15.3846	86.6162	5.3342e-08	1.9997	0.2549	-0.2263	15.3846	86.6162	1.5982	0.0940	7.6707	0.2633	-0.2122	11.7424	70.4117	1.8968	0.5288	4.8941	0.2633	-0.2122	11.7424	70.4117	1.0000	1.0000	4.0810	0.5000	-0.1154	15.3846	86.6162	0.0450	6.3723e-09	4.8005e-08	3.3877	14.2298	82.1317	0.3662	1.5365	1.1405	4.0027	0.5000	16.2500	90.1092	-0.1154	16.2500	90.1091

	a0	b0	a1	b1	a2	b2
1	18	2	7	13	14	6
2	19	1	16	4	10	10
3	11	9	7	13	10	10
4	20	0	13	7	10	10
5	16	4	6	14	9	11
6	11	9	9	11	11	9
7	13	7	7	13	12	8
8	16	4	12	8	11	9
9	20	0	16	4	14	6
10	20	0	16	4	14	6
11	11	9	8	12	6	14
12	20	0	6	14	9	11
13	20	0	8	12	12	8
14	15	5	15	5	7	13
⋮

	SubjectID	a0	b0	a1	b1	a2	b2	Chose1	Prob1
1	101	16	5	14	7	8	13	1	0
2	101	16	1	5	12	15	2	0	0
3	101	16	3	15	4	5	14	0	0
4	101	5	16	11	10	7	14	0	0
5	101	15	2	8	9	13	4	0	0
6	101	5	16	7	14	12	9	1	0
7	101	1	15	12	4	2	14	0	0
8	101	2	15	10	7	4	13	0	0
9	101	15	3	8	10	14	4	0	0
10	101	3	15	10	8	5	13	0	0
11	101	15	1	6	10	14	2	0	0
12	101	3	15	11	7	4	14	1	0
13	101	15	3	11	7	8	10	0	0
14	101	14	2	13	3	4	12	0	0
15	101	15	2	7	10	14	3	0	0
16	101	3	16	12	7	4	15	0	0
17	101	15	2	11	6	7	10	0	0
18	101	2	16	5	13	11	7	0	0
19	101	2	16	4	14	12	6	1	0
20	101	1	16	6	11	9	8	1	0
21	101	3	16	5	14	12	7	1	0
22	101	16	2	13	5	6	12	0	0
23	101	15	3	14	4	5	13	1	0
24	101	3	16	11	8	5	14	0	0
25	101	3	15	4	14	12	6	1	0
26	101	3	16	6	13	11	8	1	0
27	101	16	3	9	10	14	5	0	0
28	101	15	2	12	5	6	11	1	0
29	101	1	15	9	7	4	12	0	0
30	101	15	1	7	9	13	3	1	0
31	101	2	16	13	5	3	15	1	0
32	101	2	15	3	14	12	5	0	0
33	101	16	5	15	6	7	14	0	0
34	101	15	3	12	6	7	11	0	0
35	101	16	2	7	11	15	3	1	0
36	101	5	16	6	15	13	8	0	0
37	101	15	1	14	2	3	13	0	0
38	101	13	2	7	8	12	3	0	0
39	101	4	16	7	13	11	9	1	0
40	101	2	16	11	7	4	14	0	0
41	101	16	2	11	7	8	10	1	0
42	101	1	15	11	5	2	14	0	0
43	101	16	3	8	11	15	4	1	0
44	101	3	16	10	9	6	13	0	0
45	101	2	14	9	7	4	12	0	0
46	101	2	14	10	6	3	13	1	0
47	101	16	2	15	3	4	14	1	0
48	101	3	16	4	15	13	6	1	0
49	101	5	16	12	9	6	15	1	0
50	101	3	15	5	13	11	7	0	0
51	101	16	5	10	11	15	6	1	0
52	101	16	1	7	10	14	3	1	0
53	101	2	15	4	13	11	6	0	0
54	101	1	16	13	4	2	15	0	0
55	101	16	2	8	10	14	4	1	0
56	101	2	16	12	6	3	15	1	0
57	101	15	3	13	5	6	12	0	0
58	101	2	16	6	12	10	8	1	0
59	101	16	2	12	6	7	11	1	0
60	101	16	5	13	8	9	12	1	0
61	101	15	2	14	3	4	13	0	0
62	101	16	1	6	11	15	2	1	0
63	101	14	2	7	9	13	3	0	0
64	101	1	15	3	13	11	5	1	0
65	101	15	2	13	4	5	12	0	0
66	101	2	15	11	6	3	14	1	0
67	102	16	5	14	7	8	13	0	0
68	102	16	1	5	12	15	2	0	0
69	102	16	3	15	4	5	14	1	0
70	102	5	16	11	10	7	14	0	0
71	102	15	2	8	9	13	4	0	0
72	102	5	16	7	14	12	9	0	0
73	102	1	15	12	4	2	14	1	0
74	102	2	15	10	7	4	13	0	0
75	102	15	3	8	10	14	4	1	0
76	102	3	15	10	8	5	13	1	0
77	102	15	1	6	10	14	2	1	0
78	102	3	15	11	7	4	14	1	0
79	102	15	3	11	7	8	10	0	0
80	102	14	2	13	3	4	12	0	0
81	102	15	2	7	10	14	3	1	0
82	102	3	16	12	7	4	15	1	0
83	102	15	2	11	6	7	10	0	0
84	102	2	16	5	13	11	7	0	0
85	102	2	16	4	14	12	6	0	0
86	102	1	16	6	11	9	8	0	0
87	102	3	16	5	14	12	7	1	0
88	102	16	2	13	5	6	12	0	0
89	102	15	3	14	4	5	13	0	0
90	102	3	16	11	8	5	14	1	0
91	102	3	15	4	14	12	6	0	0
92	102	3	16	6	13	11	8	0	0
93	102	16	3	9	10	14	5	1	0
94	102	15	2	12	5	6	11	0	0
95	102	1	15	9	7	4	12	1	0
96	102	15	1	7	9	13	3	1	0
97	102	2	16	13	5	3	15	1	0
98	102	2	15	3	14	12	5	0	0
99	102	16	5	15	6	7	14	0	0
100	102	15	3	12	6	7	11	0	0
⋮

	a0	b0	a1	b1	a2	b2
1	18	2	7	13	14	6
2	19	1	16	4	10	10
3	11	9	7	13	10	10
4	20	0	13	7	10	10
5	16	4	6	14	9	11
6	11	9	9	11	11	9
7	13	7	7	13	12	8
8	16	4	12	8	11	9
9	20	0	16	4	14	6
10	20	0	16	4	14	6
11	11	9	8	12	6	14
12	20	0	6	14	9	11
13	20	0	8	12	12	8
14	15	5	15	5	7	13
⋮

	SubjectID	a0	b0	a1	b1	a2	b2	Chose1	Prob1
1	101	16	5	14	7	8	13	1	0
2	101	16	1	5	12	15	2	0	0
3	101	16	3	15	4	5	14	0	0
4	101	5	16	11	10	7	14	0	0
5	101	15	2	8	9	13	4	0	0
6	101	5	16	7	14	12	9	1	0
7	101	1	15	12	4	2	14	0	0
8	101	2	15	10	7	4	13	0	0
9	101	15	3	8	10	14	4	0	0
10	101	3	15	10	8	5	13	0	0
11	101	15	1	6	10	14	2	0	0
12	101	3	15	11	7	4	14	1	0
13	101	15	3	11	7	8	10	0	0
14	101	14	2	13	3	4	12	0	0
15	101	15	2	7	10	14	3	0	0
16	101	3	16	12	7	4	15	0	0
17	101	15	2	11	6	7	10	0	0
18	101	2	16	5	13	11	7	0	0
19	101	2	16	4	14	12	6	1	0
20	101	1	16	6	11	9	8	1	0
21	101	3	16	5	14	12	7	1	0
22	101	16	2	13	5	6	12	0	0
23	101	15	3	14	4	5	13	1	0
24	101	3	16	11	8	5	14	0	0
25	101	3	15	4	14	12	6	1	0
26	101	3	16	6	13	11	8	1	0
27	101	16	3	9	10	14	5	0	0
28	101	15	2	12	5	6	11	1	0
29	101	1	15	9	7	4	12	0	0
30	101	15	1	7	9	13	3	1	0
31	101	2	16	13	5	3	15	1	0
32	101	2	15	3	14	12	5	0	0
33	101	16	5	15	6	7	14	0	0
34	101	15	3	12	6	7	11	0	0
35	101	16	2	7	11	15	3	1	0
36	101	5	16	6	15	13	8	0	0
37	101	15	1	14	2	3	13	0	0
38	101	13	2	7	8	12	3	0	0
39	101	4	16	7	13	11	9	1	0
40	101	2	16	11	7	4	14	0	0
41	101	16	2	11	7	8	10	1	0
42	101	1	15	11	5	2	14	0	0
43	101	16	3	8	11	15	4	1	0
44	101	3	16	10	9	6	13	0	0
45	101	2	14	9	7	4	12	0	0
46	101	2	14	10	6	3	13	1	0
47	101	16	2	15	3	4	14	1	0
48	101	3	16	4	15	13	6	1	0
49	101	5	16	12	9	6	15	1	0
50	101	3	15	5	13	11	7	0	0
51	101	16	5	10	11	15	6	1	0
52	101	16	1	7	10	14	3	1	0
53	101	2	15	4	13	11	6	0	0
54	101	1	16	13	4	2	15	0	0
55	101	16	2	8	10	14	4	1	0
56	101	2	16	12	6	3	15	1	0
57	101	15	3	13	5	6	12	0	0
58	101	2	16	6	12	10	8	1	0
59	101	16	2	12	6	7	11	1	0
60	101	16	5	13	8	9	12	1	0
61	101	15	2	14	3	4	13	0	0
62	101	16	1	6	11	15	2	1	0
63	101	14	2	7	9	13	3	0	0
64	101	1	15	3	13	11	5	1	0
65	101	15	2	13	4	5	12	0	0
66	101	2	15	11	6	3	14	1	0
67	102	16	5	14	7	8	13	0	0
68	102	16	1	5	12	15	2	0	0
69	102	16	3	15	4	5	14	1	0
70	102	5	16	11	10	7	14	0	0
71	102	15	2	8	9	13	4	0	0
72	102	5	16	7	14	12	9	0	0
73	102	1	15	12	4	2	14	1	0
74	102	2	15	10	7	4	13	0	0
75	102	15	3	8	10	14	4	1	0
76	102	3	15	10	8	5	13	1	0
77	102	15	1	6	10	14	2	1	0
78	102	3	15	11	7	4	14	1	0
79	102	15	3	11	7	8	10	0	0
80	102	14	2	13	3	4	12	0	0
81	102	15	2	7	10	14	3	1	0
82	102	3	16	12	7	4	15	1	0
83	102	15	2	11	6	7	10	0	0
84	102	2	16	5	13	11	7	0	0
85	102	2	16	4	14	12	6	0	0
86	102	1	16	6	11	9	8	0	0
87	102	3	16	5	14	12	7	1	0
88	102	16	2	13	5	6	12	0	0
89	102	15	3	14	4	5	13	0	0
90	102	3	16	11	8	5	14	1	0
91	102	3	15	4	14	12	6	0	0
92	102	3	16	6	13	11	8	0	0
93	102	16	3	9	10	14	5	1	0
94	102	15	2	12	5	6	11	0	0
95	102	1	15	9	7	4	12	1	0
96	102	15	1	7	9	13	3	1	0
97	102	2	16	13	5	3	15	1	0
98	102	2	15	3	14	12	5	0	0
99	102	16	5	15	6	7	14	0	0
100	102	15	3	12	6	7	11	0	0
⋮